Data-Driven Prediction of Surface Transport Quantities in Williamson Nanofluid Flow via Hybrid Numerical Neural Approach

Abstract

This study introduces an efficient and accurate two-stage explicit computational scheme for solving partial differential equations (PDEs) containing first-order time derivatives. The suggested method is a modification of the classical Runge–Kutta scheme that introduces a new first-stage formulation. This minimizes numerical error with moderate step sizes while preserving the stability region of the classical method. Spatial discretization is performed using a sixth-order compact finite-difference scheme to obtain high-resolution solutions. The analysis of stability and convergence is strictly determined for both scalar and system forms of convection–diffusion-type equations. To illustrate the suitability of the method, a dimensionless mathematical model of the unsteady, incompressible, laminar flow of a Prandtl-type non-Newtonian nanofluid over a Riga plate is considered, accounting for viscous dissipation, thermophoresis, Brownian motion, and a magnetic field. Here, the Prandtl ternary nanofluid is defined as a non-Newtonian nanofluid that follows the Prandtl rheological model, and it exhibits three critical transport phenomena: heat conduction, viscous dissipation, and nanoparticle diffusion. Representative values of the Prandtl number $P_r=3$ and Reynolds number $R_e=5$ are used to perform the simulation, and other parameters, including but not limited to the Hartmann number $H_a$, Williamson number $W_e$, thermophoresis $N_t$ and Brownian motion $N_b$, are varied to evaluate the flow behavior. Moreover, an artificial neural network (ANN)-developed surrogate model is used to calculate the skin friction coefficient and the local Sherwood number, using five input parameters: the Reynolds number, Prandtl number, Schmidt number, Brownian motion parameter, and thermophoresis parameter. The governing partial differential equations yield high-fidelity numerical data used to train the surrogate model. The data is split into 80% for training, 10% for validation, and 10% for testing. The ANN is tested using regression analysis and error histograms, which demonstrate high accuracy and generalization capacity. Numerical simulation combined with AI-based prediction is a cost-efficient method for real-time estimation of complex non-Newtonian nanofluid systems.

1. Introduction

Nonlinear, coupled, and parameter-sensitive time-dependent partial differential equations (PDEs) occur frequently in fluid dynamics, heat transfer, and mass transport. The correct numerical solution is a key problem in computational mathematics, especially when convection–diffusion and non-Newtonian rheology are solved simultaneously. Stability, convergence, and computational efficiency must be well-balanced in such systems. Runge–Kutta (RK) methods with explicit time-integration schemes have been widely applied due to their simplicity and ease of implementation. But in nonlinear convection–diffusion problems, classical RK methods can exhibit severe stability limits and greater numerical error, especially when combined with high-resolution spatial discretization.

To overcome these limitations, modified and multi-stage explicit schemes are proposed; the schemes aim to improve stability behavior and reduce truncation error, at the cost of computational efficiency. Such time integrators, together with compact finite difference schemes that offer high-order spatial accuracy with a very narrow stencil, constitute an effective toolkit for the numerical solution of transport-dominated PDEs in mathematical fluid dynamics.

At the same time, greater attention has been paid to modeling non-Newtonian nanofluid flows, which exhibit high thermal efficiency and complex rheological behavior. The Williamson fluid model particularly suits non-Newtonian models in defining shear-thinning behavior and does not yield to stress. When nanoparticles are suspended in such fluids, the resulting Williamson nanofluid exhibits better heat and mass transfer properties due to Brownian motion and thermophoresis. The governing equations describing unsteady Williamson nanofluid flow under magnetic and thermal effects are nonlinear convection–diffusion-type PDE systems, posing significant numerical challenges.

Kumar et al. [1] examined the flow of an unsteady magnetohydrodynamic (MHD) nanofluid over a vertical plate with porosity, thermal radiation, and heat generation, using the finite difference method, similarity transformations, and the Runge–Kutta method. Moreover, topological indices provide information on the structure and reactivity of chemicals; Nadeem et al. [2] used mathematical chemistry, specifically algebraic graph techniques, to evaluate topological indices. The modeling of complex events in nanofluid transport is indirectly supported by these indices, which are crucial for comprehending the thermophysical properties of nanomaterials. Abdal et al. [3] examined the flow of an Ellis nanofluid on a stretched surface under the influence of buoyancy, magnetic fields, and heat generation. Nandeppanavar and Jagadevappa [4] employed the BVP4C solver in MATLAB R2024b to investigate heat and mass transfer during nanofluid flows of swimming microorganisms in non-Newtonian and Newtonian regimes. The authors also examined nonlinear radiation, Joule heating, viscous dissipation, and similar phenomena. Benaissa et al. [5] highlighted the effects of thermal radiation and Joule heating, using the Keller box method to investigate power-law nanofluid flow over a vertically stretched sheet. Chemical graph theorists are also increasingly adopting the approaches of algebraic graph theorists, such as those of Nadeem et al. [2], to elucidate the behavior of molecular connectivity and reactions in terms of quasi-divisor graphs. By providing structural indicators and transport characteristics, these results advance nanofluid modeling. Ahmad et al. [6] found that magneto-Sutterby nanofluids in Darcy–Forchheimer porous media could transfer heat via radiation and Brownian motion. In contrast, Israr Ur Rehman et al. [7] demonstrated that RSM was highly sensitive to thermal radiation when employed to improve the movement of Williamson nanofluid in inclined stretching cylinders.

Recent advances in heat transfer have placed non-Newtonian hybrid nanofluids at the center stage due to their high thermal conductivity and rheological properties. They exhibit shear-thinning, viscoelasticity, and yield stress, which, compared with those of Newtonian fluids, enable more accurate modeling of a broad range of biological and industrial processes. Thermodynamic principles further enhance nanoparticles’ thermal performance, stability, and energy transfer. Flows with stretched surfaces, MHD boundary layers, and porous media are especially well-suited for these systems. At stagnation points across expanding porous sheets, radiative heat and mass transfer in Maxwell, micropolar, and Williamson nanofluids is significantly enhanced by suction, especially in micropolar nanofluids, as shown numerically by Nabway et al. [8]. Zeeshan et al. [9] utilized artificial neural networks to analyze Eyring–Powell hybrid nanofluid flows across porous cylinders under MHD and heat absorption effects, and they did the same thing here. Nonlinear radiation and MHD effects were considered in the study of ternary hybrid nanofluids between spinning discs conducted by Molla et al. [10].

On the other hand, Galal et al. [11] investigated the energy transmission and entropy production in solar thermal systems using the Cattaneo–Christov heat flux model. PV [12] studied chemical reactions, and Stefan blowing ternary nanofluid flows across nonlinear stretching surfaces. Graish et al. [13] used machine learning methods to estimate nanofluid viscosity. Also, in the context of the detailed transport networks of porous and nanofluid systems, Hazzabi et al. [14] examined the combinatorial and structural properties of algebraic quasigroup representations using Latin squares and edge labeling. Moreover, Ahmad et al. [15] demonstrated that the Cattaneo–Christov double-diffusion model is better than the classical Fourier–Fick laws in finite-speed thermal and solutal diffusion of nonlinear convection and chemistry in Sutterby nanofluids.

The suspension of nanoparticles can enhance thermal conductivity, as illustrated by Choi [16], and this development was a breakthrough in nanofluids. This observation has improved research on the dynamics of nanofluids, heat transfer, and their potential industrial applications. The ongoing events have considerably enlightened us about such complex fluids. Rashid et al. [17] reported that nanoparticle shape can determine magnetohydrodynamic flow properties along long surfaces and the generation of entropy. Alali and Megahed [18] investigated the effects of velocity slip on unstable Casson nanofluid flows in the presence of heat radiation and viscous dissipation. Abbas et al. [19] investigated the transport of chemically reactive species in numerical calculations of magnetized nanofluid flow around cones at high speeds. Amer et al. [20] investigated dissipative non-Newtonian nanofluid flow in porous media with surface roughness to address real-world engineering applications.

Conversely, Garvandha et al. [21] studied thermodynamic irreversibilities of inclined magnetic nanofluid flows on a cylindrical geometry. Recently, our understanding of nanofluid behavior has advanced through studies of more complex physical processes in modern thermal applications. Ali et al. [22] investigated the effects of magnetism, nanoparticle volume fraction dynamics, and Coriolis forces on entropy generation in mixed convection to better understand thermodynamic irreversibility in rotating nanofluid environments. Mishra and Pathak [23] went a step further by studying the thermal-flow characteristics of an expanding and contracting cylinder holding a hybrid nanofluid of polytetrafluoroethylene and single-walled carbon nanotubes. They argued that slip effects are crucial for enhancing heat transfer. Mishra [24] has gone one step further to examine the radiative Ellis hybrid nanofluid flow in a slip-permeable medium and has found that radiation and porosity influence the temperature response of hybrid suspensions. Collectively, these studies contribute to understanding the effects of multiphysics interactions on momentum and heat transfer and broaden the nanofluid transport modeling paradigm. The main objective of the current research is to develop advanced modeling tools that effectively describe the coupled thermal, electromagnetic, and rheological behavior of nanofluids. All these investigations make this a necessity.

Fluid flow systems also depend on thermal properties, which vary with temperature, as shown by several seminal studies. Hamad et al. [25] investigated radiative magnetohydrodynamic stagnation-point flow and concluded that dissimilar viscosity and thermal conductivity can significantly influence the flow behavior compared with constant-property assumptions. Khader and Megahed [26] employed the differential transformation method to analyze the flow through porous media with various sheet thicknesses and different thermal conductivities, and to measure the growth of the thermal boundary layer. In their study, Hamid and Khan [27] found that variations in thermal conductivity also have a considerable impact on the formation of a boundary layer in unstable Williamson nanofluid flows. The heat transfer’s sensitivity to surface conditions is amplified by thermal conductivity gradients, according to recent research on hybrid nanofluids by Jan et al. [28]. This work was extended by Khaleque et al. [29], who examined the coupled effects of viscosity and thermal conductivity in power-law fluids undergoing radiative and diffusive transport. These innovative studies were used to develop our approach for accurately describing heat transfer dynamics in complex nanofluid regimes under the influence of coupled thermal and electromagnetic fields, including variations in thermal conductivity.

Velocity, temperature, and concentration fields are not the only interesting quantities in engineering practice. These quantities are critical for drag estimation, lubrication processes, coating technologies, chemical transport systems, and thermal management devices. Traditionally, they are computed through repeated numerical simulations, which become computationally expensive when extensive parametric studies or optimization tasks are required.

Artificial intelligence is currently founded on ANN applications. Specifically, ANNs can be distinguished by their ability to learn and restore data from nonlinear events. Given their versatility, young researchers are becoming interested in and involved with them. Numerous fields make use of artificial neural networks (ANNs), including but not limited to quantum chemistry, machine translation, finance, visualization, medical diagnosis, sequence recognition, process control, data mining, pattern recognition, system identification, and social network filtering. ANNs are multidimensional and can analyze incoming data as it passes through the network. The effective and practical application of stochastic numerical methods, especially in backpropagation, by ANNs is well-known. Backpropagation uses the supervised learning approach of gradient descent to reduce the likelihood of errors. Although Paul Werbos originally suggested the backpropagation method in 1974, it was not until 1989 that Rumelhart and Parker rediscovered it. When learning, several feed-forward multilayer neural networks employ backpropagation. One new strategy for training ANNs to handle fluid flow-related computational problems is the Levenberg–Marquardt (LM) backpropagation method.

Some writers have achieved steady convergence in both Newtonian and non-Newtonian fluid systems by employing a Levenberg–Marquardt backpropagating ANN known as LBMBN. Ly et al. [30] conducted a metaheuristic analysis of LBM-BN’s structure and specifications to determine how accurately it predicts the shear capacity of foamed concrete in a short time. Reinforced concrete beam defects were investigated by Zhao et al. [31] using the LBM-BN technique. Nguyen et al. [32] studied the application of ANN-based linguistic models (LM) to enhance the precision of robot positioning. Ali et al. [33] forecast the volumetric flow rate across a steep crest weir using an ANN trained with a Language Model (LM) method.

Recently, due to a rise in ANN-based frameworks, nanofluid studies have increased to verify and predict fluid flow. Preliminary evidence suggests that ANNs can accurately forecast the transfer of heat and mass in bioconvective nanofluids over stretched sheets [34], and it was shown in [35] that, in comparison to conventional numerical models, ANN models display nonlinear transport properties. These studies show that numerical methods using ANNs are an effective alternative to the repetitive assessment of parameters and offer stronger predictive performance. This study responds to these enquiries by verifying, via NNTS in MATLAB, the accuracy of the numerical solutions to the Casson–Jeffrey hybrid nanofluid model. Unfortunately, there are currently no high-fidelity DNS/CFD datasets or experimental results accessible in the literature for the Casson–Jeffrey hybrid nanofluid model, which incorporates two stretching surfaces, nonlinear radiation, activation energy, Brownian motion, and porous medium effects. Since no prior formulation of this configuration has been found, it is not possible to directly compare the governing equations obtained in this study to experimental or benchmark data. Since the RK-Shooting numerical solution has been proven time and time again for boundary-layer nanofluid flows, it is used as a benchmark to validate the ANN prediction.

Recent developments in time-integration methods have focused on improving stability and efficiency for nonlinear PDEs. For example, Gu et al. [36] analyzed implicit–explicit Runge–Kutta schemes for reaction–diffusion systems with stochastic diffusion coefficients, while D’Afiero [37] developed embedded strong-stability-preserving Runge–Kutta methods with adaptive time stepping for shock-dominated flows. These studies highlight the ongoing effort to design robust Runge–Kutta frameworks for complex transport equations.

Research Gap: Despite the extensive literature on (i) numerical time-integration schemes, (ii) Williamson nanofluid modeling, and (iii) ANN-based prediction in fluid systems, the following gaps remain:

• A modified explicit Runge–Kutta scheme, proven to be stable and convergent, tailored for nonlinear convection–diffusion systems, has not been systematically applied to Williamson nanofluid flow.
• High-order compact spatial discretization has not been effectively coupled with such modified time integrators in this context.
• AI-based surrogate prediction of both the skin friction coefficient and the local Sherwood number for Williamson nanofluid systems remains underexplored.

To address this challenge, artificial intelligence (AI) and data-driven modeling techniques have recently emerged as efficient alternatives for predicting complex flow-dependent quantities. By learning from high-fidelity numerical or experimental datasets, AI models can establish nonlinear mappings between governing parameters and output quantities of interest. In fluid dynamics, AI-based predictors have demonstrated the potential to predict heat transfer rates, drag coefficients, and transport properties at much lower computational cost. Nevertheless, there is still little evidence of AI models applied to predict skin friction and the Sherwood number in Williamson nanofluid flows, suggesting that this issue should be investigated further.

Inspired by these factors, the current paper proposes an adapted two-stage explicit Runge–Kutta scheme for the solution of time-dependent convection–diffusion-type PDEs arising in Williamson nanofluid flow. The proposed scheme changes the initial step of the classical Runge–Kutta method, resulting in a decrease in numerical error and the same stability region as the conventional RK scheme with the same step size. Spatial discretization is performed using high-order compact finite-difference schemes to achieve higher resolution and greater accuracy. The theory of the proposed method is shown to be reliable through a rigorous stability and convergence analysis conducted for both scalar equations and systems of convection–diffusion equations.

Objective of the Study: These shortcomings inspired the current investigation, which seeks to:

• We developed a modified two-stage explicit Runge–Kutta scheme that preserves the classical stability region while reducing numerical error.
• We coupled the scheme with a high-order compact finite difference method for spatial discretization.
• We established rigorous stability and convergence analysis for scalar and coupled convection–diffusion systems.
• We applied the developed framework to an unsteady Williamson nanofluid model with magnetic, thermal, and mass transfer effects.
• We constructed an ANN-based surrogate model trained on high-fidelity numerical data to predict the skin friction coefficient and local Sherwood number efficiently.

The proposed model will fill the gap between classical numerical analysis and modern artificial intelligence, providing a powerful, reliable framework for simulating non-Newtonian nanofluid transport processes.

Justification of the Hybrid Numerical–AI Framework: It should be noted that the ANN component employed in the paper does not replace the numerical solver and does not aim to estimate the governing partial differential equations directly. Instead, the given hybrid scheme is sequential and physically consistent. A compact, modified Runge–Kutta scheme, shown to be stable and convergent, is first developed to yield high-fidelity numerical solutions of the nonlinear Williamson nanofluid PDE system. These methods provide accurate assessments of the engineering surface quantities, namely the skin friction coefficient and the local Sherwood number. Second, the ANN model is trained solely on this numerically validated dataset to develop a surrogate mapping between the controlling physical parameters and the transport quantities. Thus, an ANN is a data-driven regression model trained with a mathematically sound solver. Such a division of roles also enables the PDE model to maintain its physical fidelity while enabling rapid prediction for parametric studies, optimization, and real-time applications. In this respect, the numerical technique ensures mathematical accuracy, while the ANN improves calculation speed. The process is therefore complementary rather than overlapping, creating a coherent multilevel scheme that reconciles deterministic numerical analysis with data-driven prediction in mathematical fluid dynamics.

The remainder of the paper is organized as follows. Section 2 presents the proposed two-stage modified Runge–Kutta scheme, detailing the time discretization and the high-order compact spatial discretization. Section 3 provides a rigorous stability and convergence analysis of the scheme for coupled convection–diffusion systems. Section 4 will describe the mathematical modeling of unsteady heat and mass transfer in a Williamson nanofluid on an oscillatory plate, including the effects of thermophoresis and Brownian motion. Section 5 presents a comprehensive numerical parametric study that quantifies the effects of key physical parameters on the solutions for velocity, temperature, and nanoparticle concentration. Section 6 develops a neural network-based surrogate model to predict the skin friction coefficient and local Sherwood number, including details of network architecture, training protocol, and performance evaluation metrics. Finally, Section 7 summarizes the main findings and discusses the future directions and practical implications of the proposed numerical and AI-based modeling framework.

2. Modified Numerical Scheme

Runge–Kutta methods are well established in the literature for discretizing time-dependent terms in partial differential equations. Numerical discretization of time-dependent convection–diffusion equations using finite difference methods and Runge–Kutta time integrators has been extensively studied in the numerical analysis literature due to their effectiveness in solving transport type partial differential equations [38,39]. In this research, one of the existing Runge–Kutta methods is modified, and the modified scheme maintains the same order of accuracy as the original one. The modified method is second-order accurate in time. To construct the modified method, the whole domain $[0,L]$ is divided into finite subintervals having equal length, denoted by $\Delta y$. Also, the whole-time interval $[0,t_f]$ is divided into a finite number of equal subintervals having length $\Delta t$. The scheme is constructed for the following time-dependent convection–diffusion PDE:

$${\displaystyle \frac{𝜕f}{𝜕t}}=\alpha {\displaystyle \frac{𝜕f}{𝜕y}}+\beta {\displaystyle \frac{{𝜕}^{2}f}{𝜕y^2}}$$

(1)

where the convective term $\alpha f_y$: advection/transport along the spatial direction and diffusion term $\beta f_{yy}$: smoothing/spreading due to viscosity/thermal diffusivity/mass diffusivity. This is the mathematical “core” of many transport problems. In the real model, $f$ can represent velocity, temperature, or nanoparticle concentration, and $\alpha ,\beta$ represent the effective convection/diffusion strengths after nondimensionalization. Equation (1) is considered subject to the boundary conditions.

$$f\left(t,0\right)=f_1,f\left(t,L\right)=f_2$$

(2)

where $f_1$ and $f_2$ are functions in time.

Classical Runge–Kutta schemes and their modified variants are widely used for time integration because of their simplicity, stability properties, and well-established order conditions derived from Taylor series expansions [39].

2.1. Predictor Stage

The first stage of the proposed scheme can be written as

$${\overline{f}}_j^{n+1}=f_j^n{e}^{-\Delta t}+(1-e^{-\Delta t})\left\{{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n+f_j^n\right\}$$

(3)

Equation (3) is called the predictor stage and computes a solution at an arbitrary time level. This stage can be used to discretize the first-order time derivative term in Equation (1) using the forward Euler method. It is an explicit predictor that produces a “best-guess” at time $n+1$. It replaces the usual forward Euler first RK stage. The factor $e^{-\Delta t}$/$\left(1−e^{-\Delta t}\right)$ gives a built-in damping/weighting that behaves nicely as $\Delta t$ varies. In convection–diffusion problems, naive explicit predictors can amplify numerical oscillations. The exponential weighting acts as a controlled relaxation toward the RHS-driven update, often reducing overshoot and improving error for practical step sizes.

2.2. Corrector Stage

The second stage of the proposed scheme is expressed as

$$f_j^{n+1}=f_j^n+\Delta t\left\{a{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n+b{\left.{\displaystyle \frac{𝜕\overline{f}}{𝜕t}}\right|}_j^{n+1}\right\}$$

(4)

where $a$ and $b$ are constants to be determined by applying the Taylor series Expansion to the left-hand side of Equation (4). This resembles a standard 2-stage RK update, but this predictor $\overline{f}$ is not the classical RK stage that is the key modification.

Expanding $f_j^{n+1}$ using the Taylor series expansion, it yields

$$f_j^{n+1}=f_j^n+\Delta t{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n+{\displaystyle \frac{{\left(\Delta t\right)}^2}{2}}{\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕t^2}}\right|}_j^n+O\left({\left(\Delta t\right)}^3\right)$$

(5)

Now, substituting the first stage (3) into Equation (4), it gives

$$f_j^{n+1}=f_j^n+\Delta t\left\{a{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n+b{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n\right\}+b\Delta t(1-e^{-\Delta t}){\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕t^2}}\right|}_j^n$$

(6)

Using the Taylor series Expansion for $f_j^{n+1}$ results in the following equations.

$$f_j^n+\Delta t{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n+{\displaystyle \frac{{\left(\Delta t\right)}^2}{2}}{\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕t^2}}\right|}_j^n=f_j^n+\Delta t\left\{a{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n+b{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n\right\}+b\Delta t(1-e^{-\Delta t}){\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕t^2}}\right|}_j^n$$

(7)

Comparing the coefficients of $f_j^n,{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n$ and ${\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕t^2}}\right|}_j^n$ on both sides of Equation (7) gives

$$\left.\begin{array}{c}a+b=1\\ b\left(1-e^{-\Delta t}\right)={\displaystyle \frac{\Delta t}{2}}\end{array}\right\}$$

(8)

Solving linear Equation (8) gives the values of $a=1-b$ and $b={\displaystyle \frac{\Delta t}{2(1-e^{-\Delta t})}}$. It means this method is second-order accurate in time, because the matching terms up to $\Delta t^2$. For small $\Delta t$, use the approximation $1-e^{-\Delta t}\approx \Delta t-\Delta t^2/2$, so $b\approx {\displaystyle \frac{1}{2}}$ and $a\approx {\displaystyle \frac{1}{2}}$.

So, the scheme smoothly connects to a familiar 2nd-order RK/trapezoid-like weighting. So, the modified proposed scheme for Equation (1) can be written as

$${\overline{f}}_j^{n+1}=f_j^n{e}^{-\Delta t}+\left(1-e^{-\Delta t}\right)\left\{\alpha {\left.{\displaystyle \frac{𝜕f}{𝜕y}}\right|}_j^n+\mathrm{\beta }{\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕y^2}}\right|}_j^n+f_j^n\right\}$$

(9)

$$f_j^{n+1}=f_j^n+\Delta t\left\{a\left(\alpha {\left.{\displaystyle \frac{𝜕f}{𝜕y}}\right|}_j^n+\mathrm{\beta }{\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕y^2}}\right|}_j^n\right)+b\left(\alpha {\left.{\displaystyle \frac{𝜕\overline{f}}{𝜕y}}\right|}_j^{n+1}+\mathrm{\beta }{\left.{\displaystyle \frac{{𝜕}^2\overline{f}}{𝜕y^2}}\right|}_j^{n+1}\right)\right\}$$

(10)

The proposed scheme discretizes only time-dependent terms in Equation (1).

2.3. Spatial Discretization Using Compact High-Order Schemes

For discretizing space-dependent terms in Equation (1), a compact high-order accurate scheme is applied.

By applying a compact scheme to Equations (9) and (10) fully discretized equations can be written as

$${\overline{f}}_j^{n+1}=f_j^n{e}^{-\Delta t}+\left(1-e^{-\Delta t}\right)\left[\alpha M_1^{-1}{N}_1{f}_j^n+\beta M_2^{-1}{N}_2{f}_j^n-f_j^n\right]$$

(11)

$$f_j^{n+1}=f_j^n+\Delta t\left[a\left(\alpha M_1^{-1}{N}_1{f}_j^n+\beta M_2^{-1}{N}_2{f}_j^n\right)+b\left(\alpha M_1^{-1}{N}_1{\overline{f}}_j^{n+1}+\beta M_2^{-1}{N}_2{\overline{f}}_j^{n+1}\right)\right]$$

(12)

where the matrices $M_1$ and $M_2$ are constructed from the coefficients of ${\left.{\displaystyle \frac{𝜕f}{𝜕y}}\right|}_{j\pm 1}^n,{ \left.{\displaystyle \frac{𝜕f}{𝜕y}}\right|}_j^n$ and ${\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕y^2}}\right|}_{j\pm 1}^n,{\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕y^2}}\right|}_j^n$ and $N_1,N_2$ are constructed from the coefficients of $f_{j\pm 1}^n,f_{j\pm 2}^n,f_j^n$ of the following equations. For spatial discretization, compact finite difference schemes are particularly attractive since they achieve high-order accuracy using relatively small computational stencils while maintaining spectral-like resolution characteristics [40,41].

First derivative compact scheme:

$${\mu }_1{\left.{\displaystyle \frac{𝜕f}{𝜕y}}\right|}_{j+1}^n+{\left.{\displaystyle \frac{𝜕f}{𝜕y}}\right|}_j^n+{\mu }_1{\left.{\displaystyle \frac{𝜕f}{𝜕y}}\right|}_{j-1}^n=b_{\circ }{\displaystyle \frac{(f_{j+1}^n-f_{j-1}^n)}{2\Delta y}}+b_1{\displaystyle \frac{(f_{j+2}^n-f_{j-2}^n)}{4\Delta y}}$$

(13)

Second derivative compact scheme:

$${\mu }_2{\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕y^2}}\right|}_{j+1}^n+{\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕y^2}}\right|}_j^n+{\mu }_2{\left.{\displaystyle \frac{{𝜕}^{2}f}{𝜕y^2}}\right|}_{j-1}^n=b_2{\displaystyle \frac{(f_{j+1}^n-2f_j^n+f_{j-1}^n)}{{\left(\Delta y\right)}^2}}+b_3{\displaystyle \frac{(f_{j+2}^n-2f_j^n+f_{j-2}^n)}{4{\left(\Delta y\right)}^2}}$$

(14)

where $b_{\circ }={\displaystyle \frac{2}{3}}\left({\mu }_1+2\right),b_1={\displaystyle \frac{1}{3}}\left(1-4{\mu }_1\right),b_2={\displaystyle \frac{4}{3}}\left(1-{\mu }_2\right),b_3={\displaystyle \frac{1}{3}}(10{\mu }_2-1)$. These can be written as linear algebra forms: $M_1{f}_y=N_{1}f,M_2{f}_{yy}=N_{2}f,$ so $f_y=M_1^{-1}{N}_{1}f, f_{yy}=M_2^{-1}{N}_{2}f.$ That is exactly how the fully discrete scheme becomes (11) and (12). In Equations (13) and (14), the parameters ${\mu }_1$ and ${\mu }_2$ are free compact-scheme weighting coefficients that control the coupling between derivative values at neighboring grid points. These parameters determine the order of spatial accuracy of the compact scheme. By selecting appropriate values of ${\mu }_1$ and ${\mu }_2$, the scheme can achieve fourth- or sixth-order accuracy in space. For example, choosing ${\mu }_1={\displaystyle \frac{1}{4}}$ yields a fourth-order accurate approximation for the first derivative, while specific optimized values of ${\mu }_2={\displaystyle \frac{1}{10}}$. Similarly, we ensure fourth- or sixth-order accuracy for the second derivative. These parameters are selected to satisfy Taylor-series consistency conditions and to minimize truncation error while preserving the compact stencil structure. This kind of scheme is very beneficial for high spatial accuracy, with narrow stencils and better spectral resolution.

2.4. Physical Relevance to the Real-World Williamson Nanofluid Problem

The time-space discretization suggested is based on the physical properties of the transport structure of the Williamson nanofluid model, in which equations of momentum, energy, and the concentration of nanoparticles have convection–diffusion operators, which are coupled by nonlinear rheology and nanoparticle flux physics. Accurate evaluation of wall gradients is essential for consistent predictions of engineering quantities, such as the skin friction coefficient and the Sherwood number. The use of a modified two-stage Runge–Kutta time integrator combined with a high-order compact scheme enhances the resolution of thin boundary layers. It suppresses nonphysical oscillations in transient and magnetohydrodynamic regimes, thereby improving the fidelity of heat and mass transfer predictions in practical applications, including magnetically controlled cooling, shear-thinning process fluids, and nanoparticle transport systems.

3. Stability and Convergence Analysis

Fourier series analysis is one of the existing stability analyses for finding stability conditions of finite difference schemes. The analysis can be applied to both linear and nonlinear partial differential equations. But the nonlinear partial differential equations must be linearized before applying this stability analysis. For linear PDEs, it gives exact stability conditions; for nonlinear PDEs, it provides an estimate of the exact stability conditions. Furthermore, the stability properties of finite difference schemes for convection–diffusion equations are commonly analyzed using the von Neumann Fourier stability analysis, which has been widely applied in the numerical analysis of PDE discretization [42].

This analysis converts the difference equations into trigonometric equations. Then, stability conditions are imposed on the obtained trigonometric equations. To apply this stability analysis, consider the following transformations:

$$M_1{e}^{jiP}={\mu }_1{e}^{(j+1)iP}+e^{jiP}+{\mu }_1{e}^{(j-1)iP}$$

(15)

$$N_1{e}^{jiP}=b_{\circ }{\displaystyle \frac{(e^{\left(j+1\right)iP}-e^{(j-1)iP})}{2\Delta y}}+b_1{\displaystyle \frac{(e^{\left(j+2\right)iP}-e^{(j-2)iP})}{4\Delta y}}$$

(16)

$$M_2{e}^{ji}={\mu }_2{e}^{(j+1)iP}+e^{jiP}+{\mu }_2{e}^{(j-1)iP}$$

(17)

$$N_2{e}^{jiP}=b_2{\displaystyle \frac{(e^{\left(j+1\right)iP}-2e^{jiP}+e^{(j-1)iP})}{{\left(\Delta y\right)}^2}}+b_3{\displaystyle \frac{(e^{\left(j+2\right)iP}-2e^{jiP}+e^{(j-2)iP})}{4{\left(\Delta y\right)}^2}}$$

(18)

where $i=\sqrt{-1}$.

Transformations (15)–(18) convert Equation (11) into the following trigonometric equation.

$${\overline{f}}_j^{n+1}=f_j^n{e}^{-\Delta t}+(1-e^{-\Delta t})\left\{\alpha \left({\displaystyle \frac{2b_{\circ }isinP+b_{1}isin2P}{2\Delta y\left(2{\mu }_1\cos P+1\right)}}\right)+\beta \left({\displaystyle \frac{4b_2(cosP-1)+b_3(cos2P-1)}{4{\left(\Delta y\right)}^2\left(2{\mu }_2\cos P+1\right)}}\right)+1\right\}{f}_j^n$$

(19)

Let $c_1=e^{-\Delta t}+(1-e^{-\Delta t})\left\{\beta \left({\displaystyle \frac{4b_2(cosP-1)+b_3(cos2P-1)}{4{\left(\Delta y\right)}^2\left(2{\mu }_2\cos P+1\right)}}\right)\right\}$ and $c_2=(1-e^{-\Delta t})\left\{\alpha \left({\displaystyle \frac{2b_{\circ }isinP+b_{1}isin2P}{2\Delta y\left(2{\mu }_1\cos P+1\right)}}\right)\right\}$.

Then Equation (19) can be rewritten as

$${\overline{f}}_j^{n+1}=(c_1+i{c}_2)f_j^n$$

(20)

Now, substituting the transformations (15)–(18) into Equation (12) results in

$$\begin{array}{cc}\hfill f_j^{n+1}=f_j^n+\Delta t& \left[a\right\{\alpha \left({\displaystyle \frac{2b_{\circ }isinP+b_{1}isin2P}{2\Delta y\left(2{\mu }_1\cos P+1\right)}}\right)\hfill \\ \hfill & +\beta \left({\displaystyle \frac{4b_2(cosP-1)+b_3(cos2P-1)}{4{\left(\Delta y\right)}^2\left(2{\mu }_2\cos P+1\right)}}\right)\}{f}_j^n\hfill \\ \hfill & +b\{\alpha \left({\displaystyle \frac{2b_{\circ }isinP+b_{1}isin2P}{2\Delta y\left(2{\mu }_1\cos P+1\right)}}\right)\hfill \\ \hfill & +\beta \left({\displaystyle \frac{4b_2(cosP-1)+b_3(cos2P-1)}{4{\left(\Delta y\right)}^2\left(2{\mu }_2\cos P+1\right)}}\right)\left\}{\overline{f}}_j^{n+1}\right] \hfill \end{array}$$

(21)

Separating the real and imaginary parts in Equation (21) yields

$$f_j^{n+1}=\left(c_3+i{c}_4\right)f_j^{n}6+(c_5+i{c}_6){\overline{f}}_j^{n+1}$$

(22)

where $c_3=1+a\Delta t\beta {\displaystyle \frac{(4b_2\left(cosP-1\right)+b_3\left(cos2P-1\right))}{4{\left(\Delta y\right)}^2\left(2{\mu }_2\cos P+1\right)}}$, $c_4=a\Delta t\alpha {\displaystyle \frac{(2b_{\circ }isinP+b_{1}isin2P)}{2\Delta y\left(2{\mu }_1\cos P+1\right)}}$, $c_5=b\Delta t\beta {\displaystyle \frac{(4b_2\left(cosP-1\right)+b_3\left(cos2P-1\right))}{4{\left(\Delta y\right)}^2\left(2{\mu }_2\cos P+1\right)}}$, and $c_6=b\Delta t\alpha {\displaystyle \frac{(2b_{\circ }isinP+b_{1}isin2P)}{2\Delta y\left(2{\mu }_1\cos P+1\right)}}$.

Using the first stage (20) in the second stage (22) and simplifying gives

$$f_j^{n+1}=\left(c_7+i{c}_8\right)f_j^n$$

(23)

where $c_7=c_3+c_5{c}_1-c_2{c}_6$ and $c_8=c_4+c_1{c}_6+c_2{c}_5$.

The stability condition can be written as

$$\left|c_7+i{c}_8\right|\le 1$$

(24)

i.e.,

$$c_7^2+c_8^2\le 1$$

(25)

So, if the scheme satisfies inequality (25), it will remain stable, or a stable solution can be achieved. But if the time and space step sizes and the parameters involved in Equation (1) do not satisfy inequality (25), an unstable solution can be obtained.

3.1. Numerical Stability Verification

A representative stability comparison plot is provided in Figure 1. The figure demonstrates the magnitude of the amplification factor $\mid G\mid$ versus the time step $\Delta t$ for both the classical RK2 scheme and the proposed modified scheme. The proposed method maintains stability across the admissible time-step range while providing enhanced damping through exponential weighting in the predictor stage. The stability threshold remains consistent with the classical RK scheme, confirming that the proposed modification neither reduces the stability region nor improves numerical robustness.

3.2. Convergence of the Proposed Scheme

The stability of the scalar convection–diffusion problem was found. Now, convergence for the system of convection diffusion will be provided. The system of convection–diffusion equations can be written in vector-matrix form.

$${\displaystyle \frac{𝜕\mathit{P}}{𝜕t}}={\lambda }_1{\displaystyle \frac{𝜕\mathit{P}}{𝜕y}}+{\lambda }_2{\displaystyle \frac{{𝜕}^2\mathit{P}}{𝜕y^2}}+{\lambda }_3\mathit{P}$$

(26)

where $\mathit{P}$ is a vector and ${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$ are matrices.

In Equation (26), the vector $\mathit{P}$ represents the collection of primary dependent variables of the coupled Williamson nanofluid system. In the present application, $\mathit{P}=\left[\begin{array}{c}u\\ \theta \\ \varphi \end{array}\right],$ where $u$ denotes the dimensionless velocity, $\theta$ the dimensionless temperature, and $\varphi$ the dimensionless nanoparticle concentration. The matrices ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ arise from the linearized form of the governing nonlinear PDE system and represent the effective convection, diffusion, and reaction/source operators, respectively. Specifically:

• ${\lambda }_1$ corresponds to the convective transport coefficients (e.g., Reynolds number-dependent advection terms);
• ${\lambda }_2$ represents the diffusive transport coefficients (including viscous, thermal, and mass diffusivity contributions);
• ${\lambda }_3$ accounts for source terms such as magnetic damping (Hartmann effect), buoyancy forces (Grashof terms), and chemical reaction parameters.

Thus, Equation (26) provides a compact matrix representation of the coupled momentum-energy-concentration system used to establish convergence of the proposed scheme. The stability and convergence proofs are carried out for this generalized operator form to ensure applicability to the full Williamson nanofluid model. By applying the proposed scheme for Equation (26), the following equations can be obtained:

$${\overline{\mathit{P}}}_j^{n+1}={\mathit{P}}_j^n{e}^{-\Delta t}+(1-e^{-\Delta t})\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \mathit{P}}_j^n+{\lambda }_2{M}_2^{-1}{N}_2{ \mathit{P}}_j^n+{\lambda }_3{\mathit{P}}_j^n+{\mathit{P}}_j^n\right\}$$

(27)

$${\mathit{P}}_j^{n+1}={\mathit{P}}_j^n+\Delta t\left[a\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \mathit{P}}_j^n+{\lambda }_2{M}_2^{-1}{N}_2{ \mathit{P}}_j^n+{\lambda }_3{\mathit{P}}_j^n\right\}+b\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \overline{\mathit{P}}}_j^{n+1}+{\lambda }_2{M}_2^{-1}{N}_2{ \overline{\mathit{P}}}_j^{n+1}+{\lambda }_3{\overline{\mathit{P}}}_j^{n+1}\right\}\right]$$

(28)

Theorem 1. 

Assume that:

• The matrices ${\lambda }_1,{\lambda }_2,{\lambda }_3$ are bounded linear operators.
• The compact discretization matrices  $M_1^{-1}{N}_1$  and  $M_2^{-1}{N}_2$  are bounded in the $\parallel \cdot {\parallel }_{\infty }$-norm.
• The time step $\Delta t$  and spatial step  $\Delta y$ are chosen such that

$$\mid {\zeta }_4\mid <1,$$

where ${\zeta }_4$ is defined in Equation (37).

Then the proposed schemes (27) and (28) are convergent, and the global error satisfies

$$E^n=O(\Delta t^2)+O(\Delta y^6).$$

Proof. 

To prove this theorem, consider the exact scheme for Equation (26) as

$${\overline{\mathit{P}}}_j^{n+1}={\mathit{P}}_j^n{e}^{-\Delta t}+(1-e^{-\Delta t})\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \mathit{P}}_j^n+{\lambda }_2{M}_2^{-1}{N}_2{ \mathit{P}}_j^n+{\lambda }_3{\mathit{P}}_j^n+{\mathit{P}}_j^n\right\}$$

(29)

$${\mathit{P}}_j^{n+1}={\mathit{P}}_j^n+\Delta t\left[a\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \mathit{P}}_j^n+{\lambda }_2{M}_2^{-1}{N}_2{ \mathit{P}}_j^n+{\lambda }_3{\mathit{P}}_j^n\right\}+b\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \overline{\mathit{P}}}_j^{n+1}+{\lambda }_2{M}_2^{-1}{N}_2{ \overline{\mathit{P}}}_j^{n+1}+{\lambda }_3{\overline{\mathit{P}}}_j^{n+1}\right\}\right]$$

(30)

Subtracting the first stage of the proposed and exact scheme (29), the following error equation can be obtained, and, letting ${\mathit{p}}_j^n-{\mathit{P}}_j^n={\mathit{E}}_j^n$, it yields

$${\overline{\mathit{E}}}_j^{n+1}={\mathit{E}}_j^n{e}^{-\Delta t}+(1-e^{-\Delta t})\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \mathit{E}}_j^n+{\lambda }_2{M}_2^{-1}{N}_2{ \mathit{E}}_j^n+{\lambda }_3{\mathit{E}}_j^n\right\}$$

(31)

By applying the norm ${‖·‖}_{\infty }$ on both sides of Equation (31), it results in

$${\overline{E}}^{n+1}\le E^n{e}^{-\Delta t}+(1-e^{-\Delta t})\left\{{‖{\lambda }_1{M}_1^{-1}{N}_1‖}_{\infty }{E}^n+{‖{\lambda }_2{M}_2^{-1}{N}_2‖}_{\infty }{E}^n+{‖{\lambda }_3‖}_{\infty }{E}^n\right\}$$

(32)

Inequality (32) can be written as

$${\overline{E}}^{n+1}\le {\zeta }_1{E}^n$$

(33)

where ${\zeta }_1=e^{-\Delta t}+(1-e^{-\Delta t})\left\{{‖{\lambda }_1{M}_1^{-1}{N}_1‖}_{\infty }+{‖{\lambda }_2{M}_2^{-1}{N}_2‖}_{\infty }+{‖{\lambda }_3‖}_{\infty }\right\}$.

Subtracting the second stage of the proposed scheme (28) and the exact scheme (30) results in

$${\mathit{E}}_j^{n+1}={\mathit{E}}_j^n+\Delta t\left[a\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \mathit{E}}_j^n+{\lambda }_2{M}_2^{-1}{N}_2{ \mathit{E}}_j^n+{\lambda }_3{\mathit{E}}_j^n\right\}+b\left\{{\lambda }_1{M}_1^{-1}{N}_1{ \overline{\mathit{E}}}_j^{n+1}+{\lambda }_2{M}_2^{-1}{N}_2{ \overline{\mathit{E}}}_j^{n+1}+{\lambda }_3{\overline{\mathit{E}}}_j^{n+1}\right\}\right]$$

(34)

Applying norm ${‖·‖}_{\infty }$ on both sides of Equation (34), it yields

$$\begin{array}{cc}\hfill E^{n+1}\le E^n& +\Delta t[a\left\{{‖{\lambda }_1{M}_1^{-1}{N}_1‖}_{\infty }{E}^n+{‖{\lambda }_2{M}_2^{-1}{N}_2‖}_{\infty }{E}^n+{‖{\lambda }_3‖}_{\infty }{E}^n\right\}\hfill \\ \hfill & \hspace{1em}\hspace{1em}+b\left\{{‖{\lambda }_1{M}_1^{-1}{N}_1‖}_{\infty }{\overline{E}}^{n+1}+{‖{\lambda }_2{M}_2^{-1}{N}_2‖}_{\infty }{\overline{E}}^{n+1}+{‖{\lambda }_3‖}_{\infty }{\overline{E}}^{n+1}\right\}]\hfill \end{array}$$

(35)

Inequality (35) can be written as follows:

$$E^{n+1}\le {\zeta }_2{E}^n+{\zeta }_3{\overline{E}}^{n+1}+C\left(O{\left(\Delta t\right)}^2,{\left(\Delta y\right)}^6\right)$$

(36)

where ${\zeta }_2=1+\left|a\right|\Delta t\left\{{‖{\lambda }_1{M}_1^{-1}{N}_1‖}_{\infty }+{‖{\lambda }_2{M}_2^{-1}{N}_2‖}_{\infty }+{‖{\lambda }_3‖}_{\infty }\right\}$.

The terms $O(\Delta t^2)$ and $O\left(\Delta y^6\right),$ appearing in Equation (36), arise from the local truncation errors of the temporal and spatial discretizations, respectively. The proposed modified Runge–Kutta scheme is constructed by matching Taylor series expansions up to second-order terms in time, which ensures a local truncation error of order $O(\Delta t^3)$ and hence a global temporal error of order $O(\Delta t^2)$. Similarly, the compact finite difference approximations used in Equations (13) and (14) are designed to achieve sixth-order spatial accuracy through Taylor series consistency conditions. Consequently, the local spatial truncation error is $O(\Delta y^6)$. Therefore, the combined consistency error in Equation (36) reflects the second-order temporal accuracy and sixth-order spatial accuracy of the proposed scheme. The truncation errors associated with finite difference approximations are typically obtained using Taylor series expansions around the grid points, which provide a systematic framework for determining the order of accuracy of numerical schemes [42].

Here, the constant $C$ denotes a generic positive constant independent of the discretization parameters $\Delta t$ and $\Delta y$. It may depend on the final time $t_f$, the bounds of the exact solution and its derivatives, and the norms of the coefficient matrices ${\lambda }_1,{\lambda }_2,{\lambda }_3$, but it does not depend on the time step or spatial grid size. This independence is essential to ensure that the global error estimate remains valid as $\Delta t\to 0$ and $\Delta y\to 0$.

Using Inequality (33) in Inequality (36) results in

$$E^{n+1}\le {\zeta }_2{E}^n+{\zeta }_3{\zeta }_1{E}^n+C(O{(\Delta t)}^2,{(\Delta y)}^6)={\zeta }_4{E}^n+C(O{(\Delta t)}^2,{(\Delta y)}^6)$$

(37)

where ${\zeta }_4={\zeta }_1{\zeta }_3$.

Let $n=0$ in Inequality (37), which gives

$$E^1\le {\zeta }_4{E}^0+C(O{(\Delta t)}^2,{(\Delta y)}^6)$$

(38)

Since the initial condition is exact, $E^0=0,$ Inequality (38) becomes

$$E^1\le C(O{(\Delta t)}^2,{(\Delta y)}^6)$$

(39)

Substituting $n=1$ in Inequality (37), it yields

$$E^2\le {\zeta }_4{E}^1+C(O{(\Delta t)}^2,{(\Delta y)}^6)\le ({\zeta }_4+1)C(O{(\Delta t)}^2,{(\Delta y)}^6)$$

(40)

Assuming $n=2$ in Inequality (37), that gives

$$E^3\le {\zeta }_4{E}^2+C(O{(\Delta t)}^2,{(\Delta y)}^6)\le ({\zeta }_4^2+{\zeta }_4+1)C(O{(\Delta t)}^2,{(\Delta y)}^6)$$

(41)

If this is continuous, then for finite $N$,

$$E^n\le ({\zeta }_4^{N-1}+{\zeta }_4^{N-2}+\cdots +{\zeta }_4+1)C(O{(\Delta t)}^2,{(\Delta y)}^6)\le ({\displaystyle \frac{1-{\zeta }_4^N}{1-{\zeta }_{47}}})C(O{(\Delta t)}^2,{(\Delta y)}^6)$$

(42)

Applying the limit as $N\to \infty$ in (42), the series $1+\dots +{\zeta }_4^{N-1}+{\zeta }_4+\dots$ becomes an infinite geometric series that will converge if $\left|{\zeta }_4\right|<1$. The expression for ${\zeta }_4$ contains norms, time, and space step sizes. So if the norm of the matrices is large, then the time and space step sizes should be appropriately chosen so that they satisfy the convergence inequality. □

3.3. Clarification About ${\zeta }_4$

It is important to emphasize that the condition $\mid {\zeta }_4\mid <1$ is a theoretically sufficient condition ensuring convergence and is not intended to be verified explicitly during practical computations. In numerical practice, stability and convergence are controlled by selecting appropriate time-step and spatial grid sizes according to standard CFL-type constraints and spectral radius considerations. The quantity ${\zeta }_4$ provides an analytical bound that captures the combined effects of operator norms and discretization parameters. Like classical Runge–Kutta schemes, stability is ensured by choosing $\Delta t$ and $\Delta y$ within admissible ranges determined by the problem coefficients. The proposed method does not require direct evaluation of ${\zeta }_4$; rather, it preserves the same stability region as the classical second-order Runge–Kutta method while decreasing numerical error. Thus, the theoretical condition serves as a rigorous mathematical guarantee rather than a computational burden.

4. Problem Formulation

Consider unsteady, incompressible, laminar Williamson nanofluid flow over the flat and oscillatory plates. The $x^{\mathrm{*}}$-axis is placed vertically, whereas the $y^{\mathrm{*}}$-axis is taken perpendicular to the $x^{\mathrm{*}}$-axis. The plate moves with a velocity $u_w$ and has higher temperatures and concentrations than the ambient values. A magnetic field $B_0$ is applied perpendicular to the plate. It is assumed that the temperature and concentration are higher along the plate than away from it. The temperature and concentration away from the plate are called the ambient temperature and ambient concentration, respectively. Heat and mass transfer occur via conduction/diffusion, and non-Newtonian rheology follows the Williamson model. Nanoparticle effects such as Brownian motion and thermophoresis are included. Chemical reaction affects the concentration equation. The governing three coupled PDEs for Williamson nanofluid can be written as follows:

$${\displaystyle \frac{𝜕u^{*}}{𝜕t^{*}}}=\nu {\displaystyle \frac{𝜕}{𝜕y^{*}}}\left[{\left(1+\mathrm{\Gamma }\left|{\displaystyle \frac{𝜕u^{*}}{𝜕y^{*}}}\right|\right)}^{-1}{\displaystyle \frac{𝜕u^{*}}{𝜕y^{*}}}\right]+g{\beta }_T\left(T-T_{\infty }\right)+g{\beta }_C\left(C-C_{\infty }\right)-{\displaystyle \frac{\sigma B_{\circ }^2}{\rho }}{u}^{*}$$

(43)

$${\displaystyle \frac{𝜕T}{𝜕t^{*}}}=\alpha {\displaystyle \frac{{𝜕}^{2}T}{𝜕{y^{*}}^2}}+{\displaystyle \frac{\mu }{c_p\rho }}{\left(1+\mathrm{\Gamma }{\displaystyle \frac{𝜕u^{*}}{𝜕y^{*}}}\right)}^{-1}{\left({\displaystyle \frac{𝜕u^{*}}{𝜕y^{*}}}\right)}^{2 }+\tau \left[D_B{\displaystyle \frac{𝜕C}{𝜕y^{*}}}{\displaystyle \frac{𝜕T}{𝜕y^{*}}}+{\displaystyle \frac{D_T}{T_{\infty }}}{\left({\displaystyle \frac{𝜕T}{𝜕y^{*}}}\right)}^2\right]$$

(44)

$${\displaystyle \frac{𝜕C}{𝜕t^{*} }}=D_B{\displaystyle \frac{{𝜕}^{2}C}{𝜕{y^{*}}^2}}+{\displaystyle \frac{D_T}{T_{\infty }}}{\displaystyle \frac{{𝜕}^{2}T}{𝜕{y^{*}}^2}}-k_1\left(C-C_{\infty }\right)$$

(45)

Here is the physical interpretation of these governing equations:

Momentum Equation (43) (Williamson viscous diffusion): ${\displaystyle \frac{𝜕u^{\mathrm{*}}}{𝜕t^{\mathrm{*}}}}$ represents the fluid’s unsteady acceleration. It captures time-dependent behavior such as startup flow, oscillatory plate motion, and transient magnetic forcing. Also, it is important in real systems such as pulsatile blood flow, time-varying cooling systems, and oscillating surfaces. $\nu {\displaystyle \frac{𝜕}{𝜕y^{\mathrm{*}}}}\left[{\left(1+\mathrm{\Gamma }\left|{\displaystyle \frac{𝜕u^{\mathrm{*}}}{𝜕y^{\mathrm{*}}}}\right|\right)}^{-1}{\displaystyle \frac{𝜕u^{\mathrm{*}}}{𝜕y^{\mathrm{*}}}}\right]$ is the core Williamson non-Newtonian stress term, where $\nu$ is the kinematic viscosity of the base fluid, ${\displaystyle \frac{𝜕u^{\mathrm{*}}}{𝜕y^{\mathrm{*}}}}$ is the shear rate, and $\mathrm{\Gamma }$ is the relaxation time parameter that controls non-Newtonian effects. When $\mathrm{\Gamma }=0$, the term reduces to the classical Newtonian viscous diffusion $\nu u_{y^{\mathrm{*}}{y}^{\mathrm{*}}}$. When $\mathrm{\Gamma }>0$, viscosity decreases with increasing shear rate, representing shear-thinning behavior. This is typical of polymer solutions, blood, paints, and food suspensions. Lower effective viscosity near the wall reduces drag and alters velocity gradients, directly affecting skin friction. $g{\beta }_T\left(T-T_{\mathrm{\infty }}\right)$ is the thermal buoyancy force where $g$ is the gravitational acceleration, ${\beta }_T$ is the thermal expansion coefficient, and $T-T_{\mathrm{\infty }}$ is the temperature difference between the fluid and the ambient temperature. It represents thermal natural convection. Hot fluid becomes less dense and rises along the vertical plate. It drives flow even in the absence of external forcing. $g{\beta }_C\left(C-C_{\mathrm{\infty }}\right)$ represents the Solutal buoyancy force, where ${\beta }_C$ is the solutal expansion coefficient, and $C-C_{\mathrm{\infty }}$ is the concentration difference. $-{\displaystyle \frac{\sigma B_{\circ }^2}{\rho }}{u}^{\mathrm{*}}$ is the magnetic (Lorentz) force, where $\sigma$ is the electrical conductivity, $B_0$ is the applied magnetic field strength, and $\rho$ is the fluid density. It represents magnetic damping (Lorentz force), opposing fluid motion, and reducing velocity.

Energy Equation (44) (Heat Transfer with Nanoparticle Effects): ${\displaystyle \frac{𝜕T}{𝜕t^{\mathrm{*}}}}$ is the unsteady heat storage. It describes the temporal change in temperature and is important in transient heating/cooling processes. $\alpha {\displaystyle \frac{{𝜕}^{2}T}{𝜕{y^{\mathrm{*}}}^2}}$ is the thermal conduction, where $\alpha$ is the thermal diffusivity, and it governs heat diffusion away from the heated plate. ${\displaystyle \frac{\mu }{c_p\rho }}{\left(1+\mathrm{\Gamma }{\displaystyle \frac{𝜕u^{\mathrm{*}}}{𝜕y^{\mathrm{*}}}}\right)}^{-1}{\left({\displaystyle \frac{𝜕u^{\mathrm{*}}}{𝜕y^{\mathrm{*}}}}\right)}^2$ is the viscous dissipation (non-Newtonian). It converts mechanical energy into heat due to internal friction, modified by Williamson rheology (shear-dependent viscosity). $\tau D_B{\displaystyle \frac{𝜕C}{𝜕y^{\mathrm{*}}}}{\displaystyle \frac{𝜕T}{𝜕y^{\mathrm{*}}}}$ represents the energy transport caused by random nanoparticle motion and couples the concentration and temperature fields. $\tau {\displaystyle \frac{D_T}{T_{\mathrm{\infty }}}}{\left({\displaystyle \frac{𝜕T}{𝜕y^{\mathrm{*}}}}\right)}^2$ represents nanoparticle migration due to temperature gradients and enhances heat transfer near the wall.

Nanoparticle Concentration Equation (45): ${\displaystyle \frac{𝜕C}{𝜕t^{\mathrm{*}}}}$ is the unsteady nanoparticle transport, $D_B{\displaystyle \frac{{𝜕}^{2}C}{𝜕{y^{\mathrm{*}}}^2}}$ is the Brownian diffusion, the random motion of nanoparticles, ${\displaystyle \frac{D_T}{T_{\mathrm{\infty }}}}{\displaystyle \frac{{𝜕}^{2}T}{𝜕{y^{\mathrm{*}}}^2}}$ is the thermophoretic diffusion, when nanoparticles migrate from hot to cold regions, and $-k_1\left(C-C_{\mathrm{\infty }}\right)$ represents the consumption or decay of nanoparticles.

Subject to initial and boundary conditions

$$\left.\begin{array}{c}\begin{array}{c}{u}^{*}=0, T=0, C=0 for t^{*}=0\\ u^{*}=u_w+L_S{\displaystyle \frac{𝜕u^{*}}{𝜕y^{*}}}, T=T_w, C=C_w when y^{*}=0\end{array}\\ u^{*}\to 0, T\to T_{\infty }, C\to C_{\infty } when y^{*}\to \infty \end{array}\right\}$$

(46)

where $u^{*}$ is the horizontal component of the velocity, $\mathrm{\Gamma }$ is the relaxation time parameter, $g$ is gravity, ${\beta }_T$ is the coefficient of Thermal convection, ${\beta }_C$ is the coefficient of solutal convection, $D_B$ is the Brownian motion coefficient, $D_T$ represents thermophoresis coefficients, $\alpha$ is the thermal diffusivity, $c_p$ is the specific heat capacity, $B_{\circ }$ is the strength of the magnetic field applied perpendicular to the plate, $\sigma$ is the electrical conductivity, $\rho$ is the density, $T_{\infty }$ is ambient temperature, and $k_1$ is the reaction rate parameter.

Equations (43)–(46) collectively model unsteady Williamson nanofluid flow with magnetic damping, buoyancy forces, nanoparticle transport, viscous heating, and chemical reaction effects. Each term is physically motivated and corresponds to mechanisms encountered in polymer processing, biomedical transport, nanofluid cooling, and magnetohydrodynamic applications.

Using the transformations

$$y={\displaystyle \frac{y^{*}}{L}},u={\displaystyle \frac{u^{*}}{u_w}}, t={\displaystyle \frac{u_w{t}^{*}}{L}},\theta ={\displaystyle \frac{T-T_{\infty }}{T_w-T_{\infty }}}, \varphi ={\displaystyle \frac{C-C_{\infty }}{C_w-C_{\infty }}}$$

(47)

Into governing Equations (44)–(47), the dimensionless system of equations can be written as

$${\displaystyle \frac{𝜕u}{𝜕t}}={\displaystyle \frac{1}{R_e}}{\displaystyle \frac{𝜕}{𝜕y}}\left({\displaystyle \frac{u_y}{1+W_e\left|\frac{𝜕u}{𝜕y}\right|}}\right)+{\displaystyle \frac{{Gr}_T}{R_e^2}}\theta +{\displaystyle \frac{{Gr}_C}{R_e^2}}\varphi -{\displaystyle \frac{H_a^2}{R_e}}u$$

(48)

$${\displaystyle \frac{𝜕\theta }{𝜕t}}={\displaystyle \frac{1}{P_r}}{\displaystyle \frac{1}{R_e}}{\displaystyle \frac{{𝜕}^2\theta }{𝜕y^2}}+{\displaystyle \frac{E_c}{R_e}}{\left(1+W_e\left|{\displaystyle \frac{𝜕u}{𝜕y}}\right|\right)}^{-1}{\left({\displaystyle \frac{𝜕u}{𝜕y}}\right)}^2+{\displaystyle \frac{N_b}{R_e}}{\displaystyle \frac{𝜕\varphi }{𝜕y}}{\displaystyle \frac{𝜕\theta }{𝜕y}}+{\displaystyle \frac{N_t}{R_e}}{\left({\displaystyle \frac{𝜕\theta }{𝜕y}}\right)}^2$$

(49)

$${\displaystyle \frac{𝜕\varphi }{𝜕t}}={\displaystyle \frac{1}{S_c}}{\displaystyle \frac{1}{R_e}}{\displaystyle \frac{{𝜕}^2\varphi }{𝜕y^2}}+{\displaystyle \frac{N_t}{{S_c{N}_{b}R}_e}}{\displaystyle \frac{{𝜕}^2\theta }{𝜕y^2}}-\gamma \varphi$$

(50)

Subject to the dimensionless initial and boundary conditions

$$\left.\begin{array}{c}u=0, \theta =0,\varphi =0 for t=0\\ u=1+\Lambda {\displaystyle \frac{𝜕u}{𝜕y}},\theta =1,\varphi =1 when y=0\\ u\to 0, \theta \to 0,\varphi \to 0 when y\to \infty \end{array}\right\}$$

(51)

where $H_a$ Hartman number, $W_e$ is Williamson’s number, $R_e$ is the Reynold number, ${Gr}_T$ and ${Gr}_C$ are the thermal and solutal Grashof numbers respectively, $E_c$ is the Eckert number, $P_r$ is the Prandtl number, $S_c$ is Schmidt number, $N_b$ is the Brownian motion parameter, $N_t$ is the thermophoresis parameter, $\gamma$ is the reaction rate parameter, and $\mathrm{\Lambda }$ is the velocity slip, which are given as follows:

$$\begin{gathered} H_a=B_{\circ }L\sqrt{{\displaystyle \frac{\sigma }{\rho \nu }}}, W_e={\displaystyle \frac{\mathrm{\Gamma }{u}_w}{L}}, R_e={\displaystyle \frac{L{u}_w}{\nu }}, {Gr}_T={\displaystyle \frac{L^{3}g{\beta }_T\left(T_w-T_{\infty }\right)}{{\nu }^2}}, {Gr}_C={\displaystyle \frac{g{\beta }_C\left(C_w-C_{\infty }\right)}{{\nu }^2}}, E_c={\displaystyle \frac{\nu u_{\circ }^2}{c_p(T_w-T_{\infty })}}, { P}_r \\ ={\displaystyle \frac{\nu }{\alpha }}, S_c={\displaystyle \frac{\nu }{D_B}}, N_b={\displaystyle \frac{\tau D_B\left(C_w-C_{\infty }\right)}{\nu }}, N_t={\displaystyle \frac{\tau D_T\left(T_w-T_{\infty }\right)}{\nu T_{\infty }}}, \gamma ={\displaystyle \frac{k_{1}L}{u_w}}, \mathrm{\Lambda }={\displaystyle \frac{L_s}{L}} \end{gathered}$$

The skin friction coefficient, local Nusselt, and Sherwood numbers are defined as

$$\left.\begin{array}{c}{C}_f={\displaystyle \frac{{\tau }_w}{\rho u_w^2}}\\ {N_u}_L={\displaystyle \frac{L{q}_w}{k\left(T_w-T_{\infty }\right)}}\\ {S_h}_L={\displaystyle \frac{L{j}_w}{D_B\left(C_w-C_{\infty }\right)}}\end{array}\right\}$$

(52)

where ${\tau }_w={\displaystyle \frac{\mu }{1+\mathrm{\Gamma }{\left|\frac{𝜕u^{*}}{𝜕y^{*}}\right|}_{y^{*}=0}}}{\left({\displaystyle \frac{𝜕u^{*}}{𝜕y^{*}}}\right)}_{y^{*}=0}, q_w=-k{\left.{\displaystyle \frac{𝜕T}{𝜕y^{*}}}\right|}_{y^{*}=0},j_w=-D_B{{\displaystyle \frac{𝜕C}{𝜕y^{*}}}⌋}_{y^{*}=0}-{\displaystyle \frac{D_T}{T_{\infty }}}{{\displaystyle \frac{𝜕T}{𝜕y^{*}}}⌋}_{y^{*}=0}$.

Using transformations (48), the dimensionless skin friction coefficient, local Nusselt, and Sherwood numbers are expressed as follows:

$$\left.\begin{array}{c}{R}_e{C}_f={\left(1+W_e{\left|{\displaystyle \frac{𝜕u}{𝜕y}}\right|}_{y=0}\right)}^{-1}{\left({\displaystyle \frac{𝜕u}{𝜕y}}\right)}_{y=0}\\ {N_u}_L=-{\left.{\displaystyle \frac{𝜕\theta }{𝜕y}}\right|}_{y=0}\\ {S_h}_L=-{\left.{\displaystyle \frac{𝜕\varphi }{𝜕y}}\right|}_{y=0}-{\displaystyle \frac{N_t}{N_b}}{\left.{\displaystyle \frac{𝜕\theta }{𝜕y}}\right|}_{y=0}\end{array}\right\}$$

(53)

4.1. Time Discretization with Modified Runge–Kutta Scheme

Discretization of Equations (48)–(50) using the proposed scheme is given as

Predictor Stage (for Velocity Equation):

$$\begin{array}{cc}\hfill {\overline{u}}_j^{n+1}=u_j^n{e}^{-\Delta t}& +\left(1-e^{-\Delta t}\right)\{{\displaystyle \frac{1}{R_e}}\left({\displaystyle \frac{M_2^{-1}{N}_2{ u}_j^n\left(1+W_e\left|M_1^{-1}{N}_1{ u}_j^n\right|\right)-W_e{M}_1^{-1}{N}_1{ u}_j^n\left|M_2^{-1}{N}_2{ u}_j^n\right|}{{\left(1+W_e\left|M_1^{-1}{N}_1{ u}_j^n\right|\right)}^2}}\right)+{\displaystyle \frac{{G_{\gamma }}_L}{R_e^2}}{\theta }_j^n\hfill \\ \hfill & +{\displaystyle \frac{{G_{\gamma }}_C}{R_e^2}}{\varphi }_j^n-{\displaystyle \frac{H_a^2}{R_e}}{u}_j^n+u_j^n \} \hfill \end{array}$$

(54)

Corrector Stage:

$$\begin{array}{cc}\hfill u_j^{n+1}=u_j^n +& \Delta t[a\left\{{\displaystyle \frac{M_2^{-1}{N}_2{ u}_j^n\left(1+W_e\left|M_1^{-1}{N}_1{ u}_j^n\right|\right)-W_e{M}_1^{-1}{N}_1{ u}_j^n\left|M_2^{-1}{N}_2{ u}_j^n\right|}{R_e{\left(1+W_e\left|M_1^{-1}{N}_1{ u}_j^n\right|\right)}^2}}\right\}+{\displaystyle \frac{{G_{\gamma }}_L}{R_e^2}}{\theta }_j^n+{\displaystyle \frac{{G_{\gamma }}_C}{R_e^2}}{\varphi }_j^n-{\displaystyle \frac{H_a^2}{R_e}}{u}_j^n\hfill \\ \hfill & \hspace{1em} +b\{{\displaystyle \frac{M_2^{-1}{N}_2{ \overline{u}}_j^{n+1}\left(1+W_e\left|M_1^{-1}{N}_1{ \overline{u}}_j^{n+1}\right|\right)-W_e{M}_1^{-1}{N}_1{ \overline{u}}_j^{n+1}\left|M_2^{-1}{N}_2{ \overline{u}}_j^{n+1}\right|}{R_e{\left(1+W_e\left|M_1^{-1}{N}_1{ \overline{u}}_j^{n+1}\right|\right)}^2}}+{\displaystyle \frac{{G_{\gamma }}_L}{R_e^2}}{ \overline{\theta }}_j^{n+1}\hfill \\ \hfill & \hspace{1em} +{\displaystyle \frac{{G_{\gamma }}_C}{R_e^2}}{\overline{\varphi }}_j^{n+1}-{\displaystyle \frac{H_a^2}{R_e}}{ \overline{u}}_j^{n+1}\left\}\right]\hfill \end{array}$$

(55)

The constants $a$ and $b$ are chosen to ensure second-order time accuracy.

Predictor Stage (for Energy Equation):

$$\begin{array}{cc}\hfill {\overline{\theta }}_j^{n+1}={\theta }_j^n{e}^{-\Delta t} +& \left(1-e^{-\Delta t}\right)\{{\displaystyle \frac{1}{P_r.R_e}}{M}_2^{-1}{N}_2{ \theta }_j^n+{\displaystyle \frac{E_c}{R_e}}{\displaystyle \frac{{\left(M_1^{-1}{N}_1{ u}_j^n\right)}^2}{1+W_e\left|M_1^{-1}{N}_1{ u}_j^n\right|}}+{\displaystyle \frac{N_b}{R_e}}\left(M_1^{-1}{N}_1{ \theta }_j^n\right)\left(M_1^{-1}{N}_1{ \varphi }_j^n\right)\hfill \\ \hfill & +{\displaystyle \frac{N_t}{R_e}}{\left(\left(M_1^{-1}{N}_1{ \theta }_j^n\right)\right)}^2+{\theta }_j^n \} \hfill \end{array}$$

(56)

Corrector Stage:

$$\begin{array}{cc}\hfill {\theta }_j^{n+1}={\theta }_j^n+\Delta t& \left\{a\right[{\displaystyle \frac{1}{P_r.R_e}}{M}_2^{-1}{N}_2{ \theta }_j^n+{\displaystyle \frac{E_c}{R_e}}{\displaystyle \frac{{\left(M_1^{-1}{N}_1{ u}_j^n\right)}^2}{1+W_e\left|M_1^{-1}{N}_1{ u}_j^n\right|}}+{\displaystyle \frac{N_b}{R_e}}\left(M_1^{-1}{N}_1{ \theta }_j^n\right)\left(M_1^{-1}{N}_1{ \varphi }_j^n\right)\hfill \\ \hfill & +{\displaystyle \frac{N_t}{R_e}}{\left(\left(M_1^{-1}{N}_1{ \theta }_j^n\right)\right)}^2]\hfill \\ \hfill & +b[{\displaystyle \frac{1}{P_r.R_e}}{M}_2^{-1}{N}_2{ \overline{\theta }}_j^{n+1}+{\displaystyle \frac{E_c}{R_e}}{\displaystyle \frac{{\left(M_1^{-1}{N}_1{ \overline{u}}_j^{n+1}\right)}^2}{1+W_e\left|M_1^{-1}{N}_1{ \overline{u}}_j^{n+1}\right|}}+{\displaystyle \frac{N_b}{R_e}}\left(M_1^{-1}{N}_1{ \overline{\theta }}_j^{n+1}\right)\left(M_1^{-1}{N}_1{ \overline{\varphi }}_j^{n+1}\right)\hfill \\ \hfill & +{\displaystyle \frac{N_t}{R_e}}{\left(\left(M_1^{-1}{N}_1{ \overline{\theta }}_j^{n+1}\right)\right)}^2\left]\right\}\hfill \end{array}$$

(57)

Predictor Stage (for Concentration Equation):

$${\overline{\varphi }}_j^{n+1}={\varphi }_j^n{e}^{-\Delta t}+(1-e^{-\Delta t})\left[{\displaystyle \frac{1}{ScRe}}{M}_2^{-1}{N}_2{\varphi }_j^n+{\displaystyle \frac{Nt}{NbScRe}}{M}_2^{-1}{N}_2{\theta }_j^n-\gamma {\varphi }_j^n+{\varphi }_j^n\right]$$

(58)

Corrector Stage:

$$\begin{gathered} {\varphi }_j^{n+1}={\varphi }_j^n+\Delta t\left\{a\left[{\displaystyle \frac{1}{ScRe}}{M}_2^{-1}{N}_2{\varphi }_j^n+{\displaystyle \frac{Nt}{Nb Sc Re}}{M}_2^{-1}{N}_2{\theta }_j^n-\gamma {\varphi }_j^n\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+b\left[{\displaystyle \frac{1}{Sc Re}}{M}_2^{-1}{N}_2{\overline{\varphi }}_j^{n+1}+{\displaystyle \frac{Nt}{Nb Sc Re}}{M}_2^{-1}{N}_2{\overline{\theta }}_j^{n+1}-\gamma {\overline{\varphi }}_j^{n+1}\right]\right\} \end{gathered}$$

(59)

4.2. Why This Scheme Is Powerful

The proposed numerical scheme combines the efficiency of an explicit Runge–Kutta-type time integrator with enhanced stability and accuracy through a novel predictor–corrector modification. Unlike traditional explicit methods, the scheme incorporates exponential damping in the predictor stage, allowing it to better handle stiffness and sharp gradients without sacrificing computational simplicity. Using time-stepping coefficients tuned by a Taylor series expansion, it is possible to achieve second-order time accuracy with the same stability region as any classical Runge–Kutta method. Furthermore, sampling compact high-order finite difference schemes for spatial discretization yields spectral-like resolution with a very narrow stencil width, thereby achieving much greater accuracy in resolving near-wall gradients, which are important for computing engineering quantities such as skin friction and the Sherwood number. The modular design of the scheme makes it easy to couple to nonlinear rheological models, such as the Williamson fluid, and to other physics, such as thermophoresis, Brownian motion, and MHD effects. The scheme is a highly effective and robust model for complex unsteady transport flows in non-Newtonian nanofluid systems, thanks to its adaptability, accuracy, and low computational cost.

4.3. Physical Relevance to Real-World Williamson Nanofluid Problems

The mathematical model and calculational framework currently in place are physically applicable to numerous real-life transport processes of shear-thinning nanofluids. The Williamson fluid model is a good model for describing the rheological behavior of a wide variety of real-world fluids that exhibit decreasing viscosity with increasing shear rate, including polymer solutions, biological fluids (blood), lubricants, and food suspensions. The inclusion of nanoparticle dynamics, including Brownian motion and thermophoresis, underscores the use of nanoparticles in modern nanofluid applications to enhance heat and mass transfer. The magnetic field term encompasses magnetohydrodynamic control, which is widely used in electromagnetic pumps, cooling of electronic equipment, and stabilization of flow in conductive fluids. Temperature and concentration gradient buoyancy forces are critical in modeling natural and mixed convection in solar collectors, chemical reactors, and environmental flows. Furthermore, the velocity slip condition at the surface is particularly pertinent to micro- and nanoscale systems, e.g., microfluidic devices and surfaces with coatings, where classical no-slip conditions do not hold. The suggested values of transport, such as the coefficient of the skin friction and the Sherwood number, give the straight engineering intuition on the nature of the drag and the efficiency of a mass transfer, which makes the proposed framework applicable to the biomedical transport, thermal management, polymer processing, and heat and mass transfer systems based on nanotechnology.

5. Results and Discussion

We conduct a comprehensive simulation study with the following aims:

• A numerical scheme is proposed for solving time-dependent partial differential equations, with a focus on nonlinear unsteady nanofluid flows. The scheme is structured to discretise only the temporal derivative terms, allowing it to be paired with various spatial discretization techniques.
• Spatial discretization is performed using a high-order compact finite difference scheme with a three-point stencil, achieving fourth- or sixth-order accuracy and effectively resolving near-wall gradients.
• The compact schemes are derived via Taylor series expansions and are well established for high-precision convection–diffusion problems.
• A modified explicit two-stage time integrator is proposed, in which an exponential predictor replaces the first Runge–Kutta stage to enhance temporal accuracy for stiff and nonlinear systems.
• The scheme is conditionally stable; the stability analysis confirms a stability region comparable to that of the classical Runge–Kutta method.
• Being an explicit approach, it does not use iterative solvers or matrix inversion, minimizing computational costs and implementation complexity.
• In nonlinear systems, explicit methods might be more accurate than large-step implicit methods, although explicit schemes only need small time steps to be stable.
• The proposed framework is applied to the dimensionless Williamson nanofluid model (Equations (48)–(51)), yielding velocity, temperature, and concentration profiles for a detailed parametric analysis of heat and mass transfer.

Remark 1. 

It is well known that high-order central or compact finite difference schemes may produce non-physical oscillations in strongly advection-dominated problems with very large Peclet numbers [38,43]. In such cases, stabilization techniques such as upwind discretization or flux-limiter methods are often used to control numerical oscillations.

5.1. Velocity Profile Analysis

The effect of the Hartmann number $H_a$ on the velocity profile $u\left(y\right)$ in an unsteady Williamson nanofluid flow over a flat or vibrating plate is shown in Figure 2. Fixed values of $\mathrm{\Lambda }=5$ (velocity slip), $We=0.04$ (Williamson number), $Re=5$ (Reynolds number), $N_t=N_b=0.1$ (thermophoresis and Brownian motion parameters), $G{r}_T=G{r}_C=0.1$ (thermal and solutal Grashof numbers), $E_c=0.1$ (Eckert number), $P_r=3$ (Prandtl number), $S_c=3$ (Schmidt number), and $\gamma =0.1$ (chemical reaction rate) are used to give the simulation. A significant reduction in fluid velocity is observed throughout the boundary layer as the Hartmann number increases from 0.1 to 2.4. This is because the Lorentz force from the applied magnetic field effectively acts as a resistive drag force against the flow of the electrically conducting fluid. The higher the value of $H_a$, the higher the magnetic damping, the thinner the boundary layers and the weaker the surface fluid motion. This fact is in line with the physics of magnetohydrodynamics (MHD), in which the Lorentz force minimizes momentum transport and slows the fluid. One can also tell in the plot that the velocity profile near the wall is stiffer than $H_a$, that is, the shear stress on the wall, which is directly proportional to the skin friction coefficient. This figure emphasizes the role of magnetic field strength in controlling the velocity field in MHD nanofluid flows, with applications in flow control, magnetic drug targeting, and electromagnetic cooling systems.

Figure 3 presents the influence of the Williamson number $W_e$ on the velocity profile $u\left(y\right)$ for unsteady nanofluid flow over a plate, under fixed physical parameters: $\mathrm{\Lambda }=5$ (velocity slip), $H_a=0.1$ (Hartmann number), $N_t=N_b=0.1$ (thermophoresis and Brownian motion parameters), $Re=5$ (Reynolds number), $G{r}_T=G{r}_C=0.1$ (thermal and solutal Grashof numbers), $E_c=0.1$ (Eckert number), $P_r=3$, $Sc=3$, and $\gamma =0.1$ (chemical reaction rate). The plot compares velocity profiles for three values of the Williamson number: $W_e=0.04$, $W_e=1.4$, and $W_e=4.0$. As $W_e$ increases, the velocity profile noticeably decreases throughout the boundary layer, indicating a reduction in flow momentum. This is indicative of the non-Newtonian shear-thinning of the Williamson fluid model: increased $W_e$ values imply increased nonlinearity and increased shear-thinning, i.e., the fluid’s resistance to shear deformation, effectively slowing the flow velocity. There is a decrease in momentum diffusion and a reduction in the boundary layer thickness. This is particularly significant in applications involving polymeric or biofluids, where rheological effects strongly influence flow behavior near solid boundaries. The velocity suppression effect is agreeable to physical intuition: the longer the relaxation time, $\mathrm{\Gamma }$ (embodied in $W_e$) the longer the delay of the fluid in its response to shear, the smaller will be the fluid motion. Overall, enhancing the Williamson number compromises the flow and strengthens shear-thinning, thereby minimizing wall shear stress and providing a means to manipulate transport in blood flow modeling, polymer extrusion in industry, and nanofluid-powered heat and mass transfer.

5.2. Temperature Profile Analysis

In Figure 4, the impact of a rise in the thermophoresis parameter $N_t$ on the dimensionless temperature distribution $\theta \left(y\right)$ across the boundary layer is illustrated. As it is demonstrated, the growth of $N_t$ up to 0.3 results in a significant increase in the temperature profile, particularly near the wall and in the thermal boundary layer. Thermophoresis (physically) causes a movement of nanoparticles (away) from the hot surface to cooler parts. Through this particle motion, more energy is lost out of the wall, and the effect is the retarding of the thermal diffusion process, which means that more thermal energy is retained close to the wall. As a result, local temperature gradients are reduced, and the overall temperature profile is brought higher with increased $N_t$. Such an improvement in the thermal field with the help of $N_t$ is essential in nanofluid-based heat transfer systems, where fine adjustments of the thermal layering are necessary. The results imply that the greater the thermophoresis effect, the higher the likelihood of thermal resistance in the area around the wall, which could be either positive or negative, depending on whether one wants thermal insulation or cooling. It is also seen in the plot that the thermal boundary layer becomes thicker with increasing $N_t$.

Figure 5 displays the effects of changing the parameter of Brownian motion, $N_b$, on the temperature distribution $\theta \left(y\right)$ in the boundary layer. Fixed values of all the other parameters are used to obtain the results, with $N_b$ as the only variable that is varied between 0.1 and 0.4. As indicated in the figure, the temperature profile across the boundary layer increases with an increase in $N_b$. This is because Brownian motion enhances thermal transport in the nanofluid. The higher the $N_b$, the higher the random motion of the suspended nanoparticles, the greater the interaction between the fluid particles and the suspended nanoparticles. This contact will facilitate further exchange of microscopic energy, leading to increased effective thermal conductivity of the fluid and higher temperatures. However, this increased energy distribution contributes to a thicker thermal boundary layer, as heat is spread over a wider area. The predicted thermal behavior of nanofluids can also explain the trend observed under enhanced Brownian motion and is particularly applicable to the design of efficient heat transfer systems that use nanofluids, where controlling thermal layering is important.

5.3. Concentration Profile

In Figure 6, the range of the nanoparticle concentration profile is displayed in relation to varying the Brownian motion parameter $N_b=\mathrm{0.1,0.2,0.3}$, and the other parameters remain constant. As the graph illustrates, the concentration boundary layer increases with the increase of $N_b$, and the concentration profile becomes more pronounced as the approach to the wall is closer, with the deterioration towards the free stream slower. In the case of lower $N_b=0.1$: The concentration profile will begin with a higher concentration towards the wall and will decrease at a steeper rate, which means that nanoparticles will be concentrated more around the surface. Further increasing $N_b$ to 0.3 causes the profile to flatten and the concentration distribution to become more widespread across the domain, indicating increased Brownian motion that broadens nanoparticle dispersion. Brownian motion is a contributing factor to the random motion of nanoparticles due to thermal agitation. This randomness is increased by increasing $N_b$ and this decreases the concentration gradient near the wall, as particles are more evenly distributed throughout the fluid. This reduces the local Sherwood number, indicating low surface mass transfer rates.

5.4. Different Parameter Analysis

Figure 7 shows how the Williamson number $W_e$ varies with time on the skin friction coefficient of a nanofluid flow in unsteady non-Newtonian Williamson fluid dynamics. The surface velocity assumes an oscillatory decaying profile $u_w=\mathrm{c}\mathrm{o}\mathrm{s}\left(5t\right)e^{-0.4t}$, which is a realistic unsteady boundary condition that occurs in other applications of density control in industry and biofluid physics. The smaller the value of the Williamson number, e.g., $W_e=0.04,$ the larger the amplitude oscillations of a skin friction profile, with slower decay. Physically, this can be seen as the increased elastic response of the fluid at low $W_e$, in which the fluid is strongly memory-effective and shear-thinning, and the velocity gradients are sharper near the wall. The higher the Williamson number (i.e., to 0.4 and 1.4), the lower the amplitude of oscillations in the skin friction and the wetter the profile becomes. This implies that the fluid would tend towards being Newtonian-like at high $W_e$ where the non-Newtonian elasticity is low, decreasing the effects of the wall shear. In general, Figure 6 verifies the fact that the skin friction coefficient of the Williamson nanofluid flow is extremely dependent on the relaxation properties of the fluid, and that the near-wall shear stress behavior can be successfully controlled by adjusting the Williamson number under the oscillatory driving forces.

Figure 8 shows how the local Nusselt number changes with time at various values of the Eckert number $E_c$, which reflects the impact of the viscous dissipation on the nature of heat transfer in the unsteady Williamson nanofluid flow over an oscillatory plate. According to the plot, the local Nusselt number decreases as $E_c$ does. Physically, a larger Eckert number implies greater viscous dissipation, which adds thermal energy to the fluid due to internal friction, thereby increasing the local fluid temperature. This rise in the temperature gradient within the thermal boundary layer reduces the rate of heat transfer at the surface, lowering the Nusselt number. Also, oscillatory behavior in the case of lower $E_c$ smooths out with increasing $E_c$, which implies a stabilizing influence of the accumulation of thermal energy. This phenomenon illustrates the importance of dissipative heating in thermal boundary layer modulation, especially in high-velocity or highly viscous flows, where the conversion of kinetic energy into internal energy is high.

Figure 9 illustrates how the local Sherwood number (a measure of mass transfer at the surface) is influenced by varying the thermophoresis parameter $N_t$, with other parameters fixed as shown in the caption of the Figure, and a time-dependent stretching velocity $u_w=\mathrm{c}\mathrm{o}\mathrm{s}\left(5t\right)e^{-0.4t}$. The higher the $N_t$ As the increment between 0.1 and 0.5, the higher the amplitude of oscillations in the Sherwood number that occurs close to the surface (the initial time zone). The first peak and trough values increase with thermophoresis, indicating that nanoparticle movement increases with thermal gradients. With time, the oscillations dissipate and approach an equilibrium state, although the ultimate amplitude is a little smaller with increasing $N_t$, which means that increasing thermophoretic strength in the medium decreases mass transfer to the wall in the long run. Physically, an increase in $N_t$ increases nanoparticle transport at the wall in response to a temperature gradient, thereby decreasing the surface concentration gradient and, in turn, the Sherwood number.

5.5. Contour Plot for Velocity, Temperature, and Concentration Profiles

Figure 10, Figure 11 and Figure 12 present contour plots of the velocity, temperature, and nanoparticle concentration fields for unsteady Williamson nanofluid flow under the influence of mixed convection and Riga plate actuation. These contours are generated for the parameter set $\mathrm{\Lambda }=5, Ha=0.1, Nb=0.1, Ec=5, Re=5, {Gr}_T=0.1, {Gr}_C=0.1, We=0.1, Pr=3, Sc=3, \gamma =0.1, Nt=0.5,$ with domain length $y_L= 17$ and a wall velocity function $u_w=\mathrm{c}\mathrm{o}\mathrm{s}\left(5t\right)e^{-0.4t}$. In Figure 9, the velocity contours exhibit strong near-wall oscillations and periodic wavefronts that gradually dissipate away from the wall, indicating the dominant influence of the time-oscillating Lorentz force generated by the Riga plate. The disturbance decays in the transverse direction, showcasing the effectiveness of the magnetic control near the surface. Figure 10 shows the temperature contours, in which the heat field slowly diffuses from the wall into the fluid domain. The gradual gradient and the presence of red-yellow areas that are prevalent in proximity to the wall imply high temperature areas because of viscous dissipation $E_c = 5$ and thermal conduction. The shift between warm and cold regions is consistent with conductive heat transfer in nanofluids, which depends on Brownian motion and thermophoresis. The slow temperature gradient demonstrates the damping effect on the velocity field’s oscillations, thereby enhancing the homogenization of the thermal field.

Figure 11 shows that the contours of nanoparticle concentration are relatively diffuse, with a high-concentration region developing near the wall and spreading outward. The thermophoresis $N_t= 0.5$ and Brownian motion $N_b= 0.1$ causes nanoparticles to move out of the hot area (wall) into the cooler fluid, replacing the individual red-blue lobes. The oscillatory zones around the wall also indicate that particle transport is not steady due to fluctuations in the velocity and activity of the oscillatory plates. Combining these numbers allows one to emphasize the intertwined nature of flow, thermal, and mass transfer processes in non-Newtonian nanofluid conditions and to assess the validity of the suggested numerical framework for tackling the complex spatiotemporal dynamics.

5.6. Computational Time for the Proposed Scheme

The contour plot of the computational time distribution of the developed hybrid numerical scheme to solve the system of governing Equations (48)–(51) under the impact of thermophoresis, magnetic field, viscous dissipation, and oscillatory stretching to the unsteady Williamson nanofluid flow is given in Figure 13. Simulations are done using the parameter set (see the captions to the figures). The horizontal axis is likely to use time steps or simulation time, and the vertical axis is likely to use spatial position or an internal iteration count. The color bar indicates a value gradient from blue (less time to compute) to red (more time to compute), so the areas of the domain where the scheme required more processing resources are shown in red. The uniform distribution of the domain shows that the computational load and stability of the proposed scheme are even. The contours’ flow and the lack of sharp elevations and shape arrangement point to a regular flow of time and a good grid resolution throughout the environment. The red area on the right-hand side indicates that regions of time-space might require greater computational power due to time effects, boundary layers, or nonlinear term interactions. This is why the scheme’s computational efficiency, stability, and scalability on a medium-sized grid are justified. The gradients consistently indicate that the method is robust even in the presence of exponential decay and periodic forcing. Therefore, the contour plot justifies the feasibility of applying this mixed numerical approach to complex Williamson nanofluid problems in two real-life industries and biomedical applications.

5.7. Convergence Comparison of the Proposed Scheme and Runge–Kutta Method for Stokes First Problem

Figure 14 shows a quantitative comparison between the Runge–Kutta method and the proposed computational scheme with respect to the computed error norm at a final time of simulation of $t_f=1$. The first Stokes problem is tested using the two methods under the same conditions; it is a classical unsteady fluid-dynamics problem used to test time-integration methods. The spatial discretization in both cases is carried out using a second-order central difference scheme, ensuring consistent spatial accuracy and isolating differences solely to time integration. The graph shows that both schemes exhibit decreasing error with mesh refinement (as desired in convergent methods). Still, the proposed scheme always gives smaller error norms than the classical Runge–Kutta method. The outcome highlights the new scheme’s increased accuracy in resolving transient dynamics, particularly in problems where part of the flow regime is dominated by diffusion, such as the Stokes problem. The increased accuracy of the proposed method can be explained by the refined first-stage time predictor, which is mathematically defined as

$${\overline{f}}_j^{n+1}=f_j^n{e}^{-15\Delta t}+{\displaystyle \frac{\left(1-e^{-15\Delta t}\right)}{15}}\left\{{\left.{\displaystyle \frac{𝜕f}{𝜕t}}\right|}_j^n+15f_j^n\right\}$$

(60)

It is an exponential decomposition of the decay term that integrates both the instantaneous value and the time derivative from the previous step, thereby improving the method’s ability to reproduce short-time dynamics with greater accuracy. The coefficient 15 in the exponential and derivative-weighted average indicates the problem’s physical or numerical stiffness and is used to accelerate convergence. In general, the reduction in errors caused by the proposed scheme is stronger at coarser time steps, indicating robustness and efficiency, particularly in problems where high accuracy is needed but fine discretization is too costly to compute. Such a performance advantage is necessary for real-time simulations or for extending the scheme to a nonlinear or coupled system.

Remark 2. 

The modified Runge–Kutta scheme developed in the preceding sections is employed to obtain high-fidelity numerical solutions of the governing Williamson nanofluid flow equations. The resulting simulation dataset is subsequently used to train an artificial neural network (ANN), which acts as a surrogate model for predicting the skin friction coefficient and the local Sherwood number without repeatedly solving the PDE system.

6. Neural Network-Based Surrogate Model

To effectively forecast the transport behavior of Williamson nanofluid flow, a neural network surrogate model is constructed to approximate the skin friction coefficient and the local Sherwood number. The surrogate model is designed to capture the complex nonlinear interactions among governing parameters and transport responses without resorting to repeated numerical simulations. The non-Newtonian Williamson rheology, the dynamics of nanoparticle diffusion, and the viscous dissipation are thoroughly represented and trained using high-fidelity numerical data derived from the coupled momentum, energy, and species-concentration equations. The Reynolds number, thermophoresis parameter, Brownian motion parameter, Prandtl number, and Schmidt number are input parameters that strongly influence the behavior of wall shear stress and mass transfer. The network outputs will be the skin-friction coefficient and the local Sherwood number, which are significant parameters for drag and mass transfer. Following training, the neural network surrogate can predict these transport quantities with great accuracy solely from the input parameters, without necessarily solving the transport equations governing each quantity, each with its own set of parameters. It is a numerical model that offers significant savings in computation and execution time, particularly in large parameter studies, optimization, and real-time prediction. The surrogate model is also modular and easily generalizes to other non-Newtonian nanofluid flow types, including heat and mass transfer; therefore, it is a versatile and effective technique for studying fluid transport.

6.1. Neural Network Configuration and Training Protocol

The artificial neural network (ANN) model is built using a feedforward multilayer perceptron (MLP) framework optimized for regression. The network architecture and training process are configured as follows:

• Input Layer: Five neurons corresponding to input parameters $Re,Nt,Nb,Pr,Sc$;
• Hidden Layers: Two hidden layers, each consisting of 10 neurons;
• Activation Functions: tansig (hyperbolic tangent sigmoid) for hidden layers, and purelin (linear) for output layer;
• Output Layer: Two neurons representing the predicted values of $C_f$ and $Sh$.

Training is carried out using the Levenberg–Marquardt backpropagation algorithm, known for its efficiency in nonlinear function approximation. The model is trained for up to 500 epochs, with early stopping based on validation performance to avoid overfitting.

Training methodology includes:

• Data Partitioning: 70% training, 15% validation, and 15% testing;
• Loss Function: Mean Squared Error (MSE);
• Regularization: Coefficient $\mu \in [{10}^{-8},{10}^{-7}]$ to improve generalization;
• Cross-validation: Applied to ensure robust predictive performance.

Each data point used for training corresponds to one simulation case with fixed flow parameters (e.g., $\mathrm{\Lambda }=5,Ha=0.1,Ec=5,We=0.1,\gamma =0.1,G{r}_T=0.1,G{r}_C=0.1$) while the core input parameters are varied. The resulting outputs $C_f$ and $Sh$ are used to teach the network the underlying nonlinear response of the system.

6.2. Model Performance Evaluation

To validate the surrogate model’s performance, the following metrics and tools were used:

• Root Mean Square Error (RMSE):

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(y_i^{\mathrm{pred}}−y_i^{\mathrm{true}}\right)}^2}$$

where $y_i^{\mathrm{pred}}$ and $y_i^{\mathrm{true}}$ denote the predicted and actual values, respectively.

• Visual Evaluation:

  ○

  Regression plots for both $C_f$ and $S_h$;

  ○

  Error histograms and residual scatter plots.

In the regression plots, the surrogate model shows a low RMSE and high correlation, confirming its ability to capture the nonlinear dynamics of Williamson nanofluid flow. This capability makes the model well-suited for:

• Real-time parametric studies;
• Sensitivity analysis;
• Inverse design and control applications.

The developed ANN surrogate model thus offers a computationally efficient and accurate approach for estimating surface transport quantities in non-Newtonian nanofluid systems, eliminating the need for repeated expensive numerical simulations.

6.3. Schematic Illustration of the Artificial Neural Network (ANN) for Flow Prediction

The architecture of the feedforward ANN was utilized as a surrogate model to estimate the surface transport characteristics, e.g., the skin friction coefficient $C_f$ and the local Sherwood number $S_h$ of the Williamson nanofluid flow are shown in Figure 15. The input layer is made of five neurons that represent major non-dimensional parameters affecting the flow, i.e., Reynolds number $R_e$, Prandtl number $P_r$, Schmidt number $S_c$, Brownian motion parameter $N_b$, and the thermophoresis parameter $N_t$. The two hidden layers, fully connected and consisting of 10 neurons each, form the hidden structure, with the tansig (hyperbolic tangent sigmoid) activation function to model the nonlinear relationship between the inputs and outputs. The third layer is the output layer, which has two neurons with a purelin (linear) activation function that generates the predicted values of $C_f$ and $S_h$. The high-fidelity numerical simulation data are used to train this surrogate ANN model.

6.4. Prediction of Skin Friction Coefficient and Local Sherwood Number Using ANN

Figure 16 demonstrates the prediction performance of a trained ANN model in estimating two critical surface transport metrics: the skin friction coefficient $C_f$ and the local Sherwood number $S_h$. The figure consists of two subplots: Left plot: ANN-predicted vs. true values for skin friction coefficient. Right plot: ANN-predicted vs. true values for local Sherwood number. Simulation and Input Parameters: The ANN model was trained using high-fidelity data generated by solving the governing equations for Williamson nanofluid flow over a Riga plate under the following physical parameters: $\mathrm{\Lambda }=5,Ha=0.1,Ec=5,G{r}_T=0.1,G{r}_C=0.1,We=0.1,\gamma =0.1$ The input parameters to the ANN include: the Reynolds number: $Re=50+w_1$; Prandtl number: $Pr=0.7+w_2$; Schmidt number: $Sc=0.6+w_3$; Brownian motion parameter: $Nb=0.01+w_4$; and thermophoresis parameter:

$$Nt=0.01+w_5$$

where $w_j$ for $j=\mathrm{1,2},\dots ,5$ are vectors of random perturbations ensuring generalization. The wall velocity function is time-dependent, $u_w=\mathrm{c}\mathrm{o}\mathrm{s}\left(5t\right)e^{-0.4t}$, and the computational domain height is $y_L=17$. Skin friction coefficient (left plot): The predicted values are closely aligned along the 45-degree reference line, indicating excellent agreement between the ANN model and the numerical solver. Local Sherwood number (right plot): Similarly, the predictions for the Sherwood number cluster around the ideal diagonal line with small deviations, confirming the ANN’s ability to generalize across varied input scenarios. These plots validate the ANN as an efficient surrogate model capable of replacing computationally expensive numerical solvers for rapid prediction tasks in complex non-Newtonian nanofluid flow systems with high accuracy.

6.5. Neural Network Training Performance

The development of the Mean Squared Error (MSE) during the neural network-based surrogate model’s training epochs is shown in Figure 17. The curves show the training, validation, and test datasets, and the green line shows the best validation (minimum validation MSE), which occurs around the 6th epoch. The experimental setup consists of simulations with $\Lambda =5,H_a=0.1,E_c=5,G{r}_T=0.1,G{r}_C=0.1,W_e=0.1,\gamma =0.1,$ and randomized vectors added to $R_e,P_r,S_c,N_b,$ and $N_t,$ and a large number of possibilities are simulated to improve generalization. First, the MSE decreases rapidly across all datasets, indicating rapid learning and successful error reduction. The training error keeps decreasing with the number of epochs, indicating that the network is learning from the information. However, the validation and test errors level off after some epochs and stabilize, showing no overfitting. The early stopping rule is applied when the validation loss stops decreasing, so that pre-training is not performed to the point where the model will not generalize well. The network consistently achieves low errors across all sets, demonstrating that the ANN is an effective predictor of the skin friction coefficient and Sherwood number in a changing flow and thermophysical regime of the Williamson nanofluid.

6.6. Error Histogram for Surrogate Model Prediction

Figure 18 presents the error histogram evaluating the performance of the ANN surrogate model in predicting the skin friction coefficient and the local Sherwood number under varying parameter perturbations. These perturbations are introduced via random vectors $w_j$ in key input parameters such as the Reynolds number ($Re=50+w_1$), Prandtl number ($Pr=0.7+w_2$), Schmidt number ($Sc=0.6+w_3$), Brownian motion parameter ($Nb=0.01+w_4$), and thermophoresis parameter ($Nt=0.01+w_5$), while all other governing physical parameters (e.g., $\mathrm{\Lambda }=5$, $Ha=0.1$, $Ec=5$, etc.) are held constant. In this histogram:

• The x-axis represents the range of prediction errors (i.e., differences between true and predicted values).
• The y-axis shows the number of instances (data samples) corresponding to each error bin.
• The bars are color-coded: blue for training data, green for validation data, and red for test data.
• The orange vertical line denotes the zero-error mark, i.e., perfect prediction.

The histogram reveals that:

• A majority of errors are centered around zero, indicating that the model performs with high precision.
• Most predictions fall into narrow error bins near the zero line, especially for the training data, suggesting that the ANN has effectively learned the underlying data distribution.
• The validation and test errors also show a tight distribution near zero, indicating strong generalization and minimal overfitting.
• The presence of only a few outlier bins with higher error magnitudes reflects occasional deviations, likely due to nonlinear interactions under certain random parameter combinations.

In general, this error histogram validates the accuracy, reliability, and strength of the trained ANN surrogate model in predicting the nonlinear relationship between input parameters and output physical quantities, i.e., shear stress and mass transfer rate in unsteady Williamson nanofluid flow.

6.7. Regression Performance of Neural Network Model

Figure 19 shows the regression efficiency of the trained ANN model in predicting two important physical quantities: the skin friction coefficient and the local Sherwood number for unsteady Williamson nanofluid flowing under oscillatory wall conditions. An ANN model is experimented with a dataset created by taking high-fidelity numerical simulations, whereby five input parameters $R_e$, $P_r$, $S_c$, $N_b$, and $N_t$ are perturbed by incorporating randomized perturbations $w_j$ (where $j=\mathrm{1,2},...,5$). The other physical parameters are given: $\Lambda =5, H_a=0.1, E_c=5, G{r}_T=G{r}_C=0.1, W_e=0.1, \gamma =0.1, y_L=17, and u_w=cos\left(5t\right)e^{-0.4t}$. The figure’s subplots correspond to the various dataset splits: training (top-left), validation (top-right), testing (bottom-left), and combined data (bottom-right). The black circles indicate the real (target) values of the numerical solver, and the colored regression line indicates the values predicted by the ANN. The dotted line $Y=T$ is the optimum reference line where the forecasts get the exact figures. The training regression plot (blue line) shows almost a perfect linear fit, indicating that the underlying data patterns are well learned. Nonetheless, the validation (green) and test (red) plots exhibit greater scatter and move farther from the optimal line, at least towards high magnitudes, suggesting some small underfitting or sparse data in those areas. The total regression (black line) continues to exhibit strong generalization, maintaining continuous alignment near the $Y=T$ line. This analysis verifies that the ANN model accurately predicts skin friction and the Sherwood number across a wide range of parametric variables, while also identifying areas where additional training samples could improve resilience.

The ANN architecture emerged from comparative experimentation with several architectures and different hidden-layer sizes and activation functions. To achieve the right bias-variance trade-off, validation RMSE and regression performance were used to select the model. The surrogate model is interpolative within the sampled parameter space and is not intended to extrapolate beyond the training space.

The inputs were zero-centered to avoid gradient saturation during the tansig activation function. The medium network size had guaranteed approximately steady conditioning of the Levenberg–Marquardt Hessian approximation. While the current architecture performs efficiently for moderate dataset sizes, extending it to high-dimensional parameter spaces or large-scale datasets may require alternative optimization strategies.

7. Conclusions

This research paper presented a cohesive computational and artificial intelligence model for numerical analysis and the effective prediction of transport phenomena in Williamson nanofluid flow. An explicit two-step Runge–Kutta scheme, with a modified version, was developed to solve time-dependent convection–diffusion partial differential equations. The suggested time integration scheme is a modification of the initial step of the classical Runge–Kutta scheme, resulting in a decrease in numerical error with the same stability region as the conventional scheme, provided the time-step sizes are appropriately chosen.

The suggested scheme was to be combined with a high-order compact finite-difference spatial discretization that provided higher resolution and accuracy for nonlinear transport problems. Both scaling equations and systems of convection–diffusion equations were tested for strict stability and convergence, confirming that the numerical method was reliable and robust. The comparison indicated that the proposed scheme is convergent and stable, and that its results are superior to those of the classical Runge–Kutta schemes.

It relied on a numerical model of Williamson nanofluid flow based on surface heat and mass transfer on a dimensionless basis. The resulting simulations provided an in-depth analysis of the dynamics of velocity, temperature, and nanoparticle concentration. The parametric tests also showed that, at a high Williamson number, shear-thinning behavior is severely disrupted, leading to a reduction in flow velocity. On the other hand, higher Hartmann numbers slow down the flow due to greater magnetic damping. In addition, the parameters of thermophoresis and Brownian motion were found to be essential for controlling nanoparticle concentration distribution and mass transfer.

In addition to numerical simulation, an AI-based predictive model was constructed to estimate the skin friction coefficient and the local Sherwood number using key physical input parameters, including the Reynolds number, Prandtl number, Schmidt number, thermophoresis parameter, and Brownian motion parameter. The AI model was found to be highly predictive, reflecting numerical trends at a much lower computational cost. This underscores the usefulness of data-driven methods as an efficient proxy of recurrent high-fidelity numerical simulations. The final arguments can be made as follows:

• The hybrid numerical method that uses the modified exponential integrator with a compact finite difference scheme has a superior performance to the classical Runge–Kutta method in terms of nonlinearities and stiffness in the governing equations, convergence, and computational cost.
• This proves to be crucial since machine learning combined with high-order numerical solvers can realize the real-time prediction and parametric sensitivity analysis of complex non-Newtonian nanofluid systems; this will lead to AI-based design optimization of thermal-fluid systems.
• The regression and error histogram plots demonstrate the validity of the ANN-based surrogate model, with low RMSE and close concentration around the optimal prediction line, indicating that it can be used for high-fidelity numerical simulations.
• The error plots and regression diagrams of the training, validation, and testing data sets show that the accuracy of the predictions is directly related to the diversity and the balance of training samples, which highlights the importance of the creation of representative datasets in the form of tables at the stage of surrogate model design.

In general, the findings support the supposition that the proposed modified Runge–Kutta scheme is an effective, accurate, and computationally efficient method for modeling the transport of nonlinear non-Newtonian nanofluids. The incorporation of artificial intelligence also contributes to greater practical use of the framework, enabling rapid prediction of engineering quantities of interest.

It can be anticipated that the current approach will be generalized to fractional- and fractal-time models, turbulent flow regimes, and three-dimensional configurations. Also, the introduction of experimental data to train AI models and investigate more sophisticated deep learning designs may enhance predictive precision and expand the range of proposed solutions.