Search Results (116)

Numerical stability is a critical property of ODE solvers that must be taken into account when choosing a numerical method for solving a certain initial value problem. However, conventional approaches to evaluating numerical stability are not directly applicable to some of the recently reported numerical integration schemes, i.e., semi-implicit or semi-explicit methods, due to their unique right-hand side function calculation principles and existence for systems of order 2 and higher. Furthermore, the stability regions of semi-implicit ODE solvers are found to be significantly influenced by the asymmetry of the simulated system. Thus, the development of specialized techniques is required to address these challenges. In the current study, a complex but straightforward methodology is proposed for the acquisition and graphical representation of the stability regions of semi-implicit integration methods with visualization in three-dimensional space. To cover the deficiency of the stability analysis for recently reported composition methods, we have chosen several composition ODE solvers as test methods. The obtained results supplement our understanding of these algorithms and reveal their strengths and weaknesses. Full article

This paper investigates a modified one-dimensional Moore–Gibson–Thompson (MGT) thermoelasticity model that significantly extends the classical formulation by incorporating two key structural modifications: frictional damping and a novel cross-coupling structure. The system introduces a viscous frictional damping mechanism proportional to the velocity acting on the mechanical (elastic) field, enhancing dissipation, which is a common feature in models extending Green–Naghdi Type III thermoelasticity. The core novelty, however, lies in introducing an additional coupling structure that explicitly links the thermal relaxation effects with the mechanical dissipation effects. This modification moves beyond the standard MGT coupling and is rooted in an effort to model complex visco-thermal interactions, representing the primary contribution to the literature. The well posedness of this modified system is first established using semigroup theory. Through the construction of a new Lyapunov functional, sufficient conditions are then rigorously derived, ensuring the exponential stability of solutions under specific parameter regimes. Furthermore, a critical balance condition is identified between the thermal conductivity and the thermal relaxation time, beyond which the system’s energy decay ceases to be exponential. Finally, numerical experiments employing an explicit–implicit finite difference scheme validate the theoretical findings and illustrate the substantial influence of both the modified coupling and the frictional damping on the system’s long-term energy behavior. Full article

This study presents a systematic comparative analysis of nine stator current discretization methods within the Model Predictive Current Control (MPCC) framework for Permanent Magnet Synchronous Motors (PMSMs). These methods have generally been examined individually or in limited combinations in previous research, and this holistic and comprehensive comparison constitutes the core contribution of this work by addressing a significant gap in the existing literature. The investigated MPCC methods—Forward Euler (FE), Backward Euler (BE), Midpoint Euler (ME), Fourth-Order Runge–Kutta (RK4), Runge–Kutta Ralston (RKR), Taylor Series (TS), Verlet Integration (VI), Crank–Nicolson (CN), and Adams–Bashforth (AB)—are comprehensively evaluated for their dynamic performance, including speed tracking, torque response, settling time, rise time, overshoot, and Total Harmonic Distortion (THD). Additionally, these analysis results are benchmarked against conventional Proportional–Integral–Derivative (PID) and Field-Oriented Control (FOC) methods. In terms of key performance indicators, the MPCC–RKR method proved optimal for speed tracking under no-load conditions, achieving the lowest overshoot, specifically ranging from 0.097% to 1.450%. Conversely, MPCC–ME and MPCC–CN demonstrated superior transient performance under sudden-load conditions (1.7 Nm), yielding the smallest torque deviations, fastest settling times. Specifically, MPCC-ME recorded the lowest overshoot (1.512%) at the 7 s load step, while MPCC-CN performed best at 9 s (1.220%) and 11 s (1.577%). Among the predictive schemes, the MPCC–RKR method achieved the highest current quality with a minimum THD of 3.69% at nominal speed. Finally, it has been confirmed through the applied statistical analysis techniques that the performance differences among the discretization methods are significant. The comparative analysis examines both the dynamic performance of the methods and the fundamental trade-off between accuracy and computational burden in MPCC design. Simple single-step explicit methods (FE, ME, RKR, VI, AB) offer low computational cost and are well suited for high–sampling-frequency real-time applications, especially with sufficiently small sampling times, whereas more complex multi-step or implicit methods (BE, RK4, TS, CN) may increase the processor load despite their potential gains in accuracy and stability. This study provides practical, evidence-based guidelines for selecting an optimal discretization method by balancing accuracy and dynamic performance requirements for PMSM applications. Full article

Tumor angiogenesis models based on coupled nonlinear parabolic partial differential equations require solving stiff systems where explicit time-stepping methods impose severe stability constraints on the time step size. Implicit–Explicit (IMEX) schemes relax this constraint by treating diffusion terms implicitly and reaction–chemotaxis terms explicitly, reducing each time step to a single linear system solution. However, standard Gaussian elimination with partial pivoting exhibits cubic complexity in the number of spatial grid points, dominating computational cost for realistic discretizations in the range of 400–800 grid points. This work presents a CUDA-based parallel algorithm that accelerates the IMEX scheme through GPU implementation of three core computational kernels: pivot finding via atomic operations on double-precision floating-point values, row swapping with coalesced memory access patterns, and elimination updates using optimized two-dimensional thread grids. Performance measurements on an NVIDIA H100 GPU demonstrate speedup factors, achieving speedup factors from 3.5× to 113× across spatial discretizations spanning $M\in [25,800]$ grid points relative to sequential CPU execution, approaching 94.2% of the theoretical maximum speedup predicted by Amdahl’s law. Numerical validation confirms that GPU and CPU solutions agree to within twelve digits of precision over extended time integration, with conservation properties preserved to machine precision. Performance analysis reveals that the elimination kernel accounts for nearly 90% of total execution time, justifying the focus on GPU parallelization of this component. The method enables parameter studies requiring $\sim {10}^4$ PDE solves, previously computationally prohibitive, facilitating model-driven investigation of anti-angiogenic therapy design. Full article

The initial part of this study fills a notable research gap by investigating the substantial impact of numerical diffusion errors from different schemes on sloshing tank models. Multiple numerical models were developed: first- and higher-order upwind schemes equipped with precise wall treatment using ghost nodes, MacCormack and central methods that are explicit second-order finite difference methods, and Preissmann and staggered methods employed in full-implicit and semi-implicit modes. Furthermore, the separation of variables technique was proposed for simulating sloshing tanks and deriving an analytical equation for the tank’s natural period. An analytical solution to the perturbation was employed to examine the numerical diffusion of the schemes. Subsequently, two sloshing tests, resonant and near-resonant excitations, were employed to determine the numerical diffusion and calibrate the physical diffusion coefficients, respectively. Finally, an efficient and accurate numerical scheme was applied to a linear shallow water model including physical diffusion and coupled with a single degree of freedom (SDOF), to simulate tuned liquid dampers (TLDs). It shows that the efficiency of TLD is associated with a compact domain around resonance excitation. Contrary to SDOF alone, when SDOF interacts with TLD the impact of structural damping on reducing the response is minimal in resonance excitation. Full article

Partial differential equations are an integral part of modern scientific development. Hyperbolic partial differential equations are encountered in many fields and have many applications—both linear and nonlinear types, with some being semilinear and quasilinear. In this paper, a family of implicit numerical schemes for solving hyperbolic partial differential equations is derived, utilizing finite differences and tridiagonal sweep. Through the discrete Fourier transform, a necessary and sufficient condition for convergence is proven for the linear version of the family of difference schemes, expanding the known results on boundary conditions that ensure convergence. Numerical verification confirms the found condition. A series of experiments on different boundary conditions and semilinear hyperbolic PDEs show that the same condition seems to also hold in those cases. In view of the results, an optimal subset of the family is found. A comparison between the implicit schemes and an explicit analogue is conducted, demonstrating the gained efficiency of the implicit schemes. Full article

Solving stiff partial differential equations with neural networks remains challenging due to the presence of multiple time scales and numerical instabilities that arise during training. This paper addresses these limitations by embedding the mathematical structure of implicit–explicit time integration schemes directly into neural network architectures. The proposed approach preserves the operator splitting decomposition that separates stiff linear terms from non-stiff nonlinear terms, inheriting the stability properties established for these numerical methods. We evaluate the methodology on Allen–Cahn equation dynamics, where interface evolution exhibits the multi-scale behavior characteristic of stiff systems. The structure-preserving architecture achieves improvements in solution accuracy and long-term stability compared to conventional physics-informed approaches, while maintaining proper energy dissipation throughout the evolution. Full article

In this work we propose a new second-order implicit–explicit difference scheme for a pseudoparabolic equation with a nonlinear flux term. The proposed method is evaluated in comparison with two known numerical approaches. The significance of this study stems from its relevance to physical and mechanical models, including the Benjamin–Bona–Mahony (–Burgers) equations, which arise as specific instances of the considered equation. A stability analysis of the constructed scheme is conducted. Furthermore, the method is generalized to address a two-dimensional case. Several numerical experiments are carried out to assess the accuracy and efficiency of the proposed scheme. Full article

Geophysical flows are characterized by rapid rotation. Simulating these flows requires small timesteps to achieve stability and accuracy. Numerical stability can be greatly improved by the implicit integration of the terms that are most responsible for destabilizing the numerical scheme. We have implemented an implicit treatment of the Coriolis force in a rotating spherical shell driven by a radial thermal gradient. We modified the resulting timestepping code to carry out steady-state solving via Newton’s method, which has no timestepping error. The implicit terms have the effect of preconditioning the linear systems, which can then be rapidly solved by a matrix-free Krylov method. We computed the branches of rotating waves with azimuthal wavenumbers ranging from 4 to 12. As the Ekman number (the non-dimensionalized inverse rotation rate) decreases, the flows are increasingly axially independent and localized near the inner cylinder, in keeping with well-known theoretical predictions and previous experimental and numerical results. The advantage of the implicit over the explicit treatment also increases dramatically with decreasing $\mathrm{Ek}$, reducing the cost of computation by as much as a factor of 20 for Ekman numbers of order of ${10}^{-5}$. We carried out continuation for both the Rayleigh and Ekman numbers and obtained interesting branches in which the drift velocity remained unchanged between pairs of saddle–node bifurcations. Full article

This study investigates the full penetration simulation of piles from the ground surface, focusing on frictional contact modeling without mesh distortion. To overcome issues related to mesh distortion and improve solution convergence, the Arbitrary Lagrangian–Eulerian (ALE) adaptive mesh technique was implemented within both explicit and implicit time integration schemes. The numerical model was validated against field experiments conducted at Bothkennar, Scotland, using the Imperial College instrumented displacement pile (ICP) in soft clay, where the soil behavior was effectively represented using the modified Cam-Clay model and the Mohr–Coulomb model. The primary objectives of this study are to evaluate the ALE method performance in handling mesh distortion; analyze the effects of soil–pile interface friction, pile dimensions, and various dilation angles on pile resistance; and compare the effectiveness of explicit and implicit time integration schemes in terms of stability, computational efficiency, and solution accuracy. The ALE method effectively modeled pile penetration in Bothkennar clay, validating the numerical model against field experiments. Comparative analysis revealed the explicit time integration method as more robust and computationally efficient, particularly for complex soil–pile interactions with higher friction coefficients. Full article

Composite materials are increasingly used in aerospace applications due to their high strength-to-weight ratio, but their fire safety remains a critical concern. This study develops a coupled thermochemical model to predict the thermal response and decomposition behavior of composite materials under high-temperature fire conditions. The framework integrates heat transfer, resin pyrolysis kinetics, and gas generation dynamics, employing the Rule of Mixtures to dynamically update temperature-dependent thermophysical properties (thermal conductivity, specific heat capacity, and density). Decomposition kinetics are governed by an n-th-order Arrhenius equation, explicitly resolving the gas convection effects on energy transport. The governing equations are solved numerically using a hybrid explicit/implicit finite element scheme, ensuring stability under severe thermal gradients. Experimental validation compliant with the 14 CFR Part 25 and ISO 2685 standards demonstrates high predictive accuracy. The model successfully captures key phenomena, including the char layer insulation effects, transient heat flux attenuation, and decomposition-induced property transition. This work establishes a computational foundation for optimizing fire-resistant composites in aerospace applications, addressing critical gaps in the existing models through coupled multiphysics representation. Full article

An implicit flux-corrected transport (FCT) and diffusion algorithm was developed and used in many gas discharge calculations. Such calculations require the use of a fine mesh where the electric field changes rapidly; that is, near electrodes or in a streamer front. If diffusion is included using an explicit method, then the von Neumann stability condition severely limits the time-step that can be used; however, this limitation does not apply to implicit methods. Further, for gas discharge calculations including space-charge effects, it is necessary to solve the continuity equations with no negative number densities nor point-by-point oscillation in the number density. This is because the electron number densities are finely balanced with the ion number densities to determine the space-charge distribution and hence the electric field which drives the motion of the particles. An efficient way to solve the particle transport equation, with the required properties, is to use FCT. The most accurate form of FCT developed by the author is implicit fourth-order FCT; hence, the method presented incorporates implicit diffusion into the implicit fourth-order FCT scheme to produce a robust algorithm that has been successfully used in many calculations. Full article

The modeling of the one-dimensional wave equation is the fundamental model for characterizing the behavior of vibrating strings in different physical systems. In this work, we investigate numerical solutions for the one-dimensional wave equation employing both explicit and implicit finite difference schemes. To evaluate the correctness of our numerical schemes, we perform extensive error analysis looking at the $L^1$ norm of error and relative error. We conduct thorough convergence tests as we refine the discretization resolutions to ensure that the solutions converge in the correct order of accuracy to the exact analytical solution. Using the von Neumann approach, the stability of the numerical schemes are carefully investigated so that both explicit and implicit schemes maintain the stability criteria over simulations. We test the accuracy of our numerical schemes and present a few examples. We compare the solution with the well-known spectral and finite element method. We also show theoretical proof of the stability and convergence of our numerical scheme. Full article

The numerical solution to boundary differential problems is a crucial task in modern applied mathematics. Usually, implicit integration methods are applied to solve this class of problems due to their high numerical stability and convergence. The known shortcoming of implicit algorithms is high computational costs, which can become unacceptable in the case of numerous right-hand side function calls, which are typical when solving boundary problems via the shooting method. Meanwhile, recently semi-implicit numerical integrators have gained major interest from scholars, providing an efficient trade-off between computational costs, stability, and precision. However, the application of semi-implicit methods to solving boundary problems has not been investigated in detail. In this paper, we aim to fill this gap by constructing a semi-implicit boundary problem solver and comparing the performance of explicit, semi-implicit, semi-explicit, and implicit methods using a set of linear and nonlinear test boundary problems. The novel blinking solver concept is introduced to overcome the main shortcoming of the semi-implicit schemes, namely, the low convergence on exponential solutions. The numerical stability of the blinking semi-implicit solver is investigated and compared with existing methods by plotting the stability regions. The performance plots for investigated methods are obtained as a dependence between global truncation error and estimated computation time. The experimental results confirm the assumption that semi-implicit numerical methods can significantly outperform both explicit and implicit solvers while solving boundary problems, especially in the proposed blinking modification. The results of this study can be efficiently used to create software for solving boundary problems, including partial derivative equations. Constructing semi-implicit numerical methods of higher-accuracy orders is also of interest for further research. Full article

This work considers the isothermal process of incompressible viscous fluid filtration in an oil-saturated, fractured-porous reservoir. A study of the pressure and water saturation distribution process is carried out for a case in which a production well is put into operation. For this problem, i.e., a mathematical model in a two-dimensional formulation, a numerical method and a parallel algorithm are proposed. The mathematical model of two-phase filtration is written in accordance with the classical laws of continuum mechanics and Darcy’s law and also includes a function of fluid exchange between low-permeability pores and high-permeability natural fractures within the framework of the Warren–Root model. The numerical solution is based on the finite-difference method and a splitting scheme of physical processes and spatial coordinates. For a split system with respect to piezoconductivity, an implicit finite-difference scheme with fixed saturations is constructed, and with respect to saturation transfer, explicit and implicit difference schemes are constructed. For parallel implementation of the developed numerical approach, a method based on geometric parallelism is selected. Testing of the developed method is performed using the example of calculating liquid mass transfer for a wide range of parameters. To verify the model, the obtained calculated pressure curves are compared with field data recorded by a deep-well measuring device. The results allow for estimation of the distribution of reservoir pressure and water saturation depending on the permeability of the fracture set and the pore part. The obtained results allow for monitoring of well operations, reducing unexpected accident risks and optimizing the development system in order to increase oil production in fractured-porous reservoirs. Computational experiments confirm the efficiency of the developed numerical algorithm and its parallel implementation. Full article