Search Results (111)

The durability of reinforced concrete structures in aggressive environments is strongly influenced by the ingress of chloride and other harmful ions, which is further complicated under partially saturated conditions, due to the coexistence of liquid and gas phases within the pore network. This study aimed to develop a predictive moisture–chemical–temperature model and to elucidate the mechanisms governing ion transport in partially saturated concrete. A multi-species hygro-chemo-thermo transport model was formulated based on the Nernst–Planck equation, incorporating electroneutrality, zero current conditions, and the coupled effects of moisture and temperature gradients. The model was numerically implemented using a parallel FE method with the Crank–Nicolson scheme, supported by domain decomposition and SPMD techniques for high computational efficiency. As a result, experimental validation was performed through chloride ponding tests under varying temperature conditions (20 °C, 35 °C, 50 °C), water-to-cement ratios (0.55, 0.65), and relative humidity differences (100%, 60%). The simulation results showed good agreement with the experimental data and confirmed that the proposed model can effectively predict chloride penetration under both isothermal and non-isothermal conditions. Additionally, the simulations revealed that moisture gradients accelerate ion transport, as the inward migration of the moisture front enhances the diffusion rates of chloride, sodium, and calcium ions until a steady-state moisture distribution is reached. Full article

To understand phase-transition processes like solidification, phase-field models are frequently employed. In this paper, we study a finite-difference Crank–Nicolson–ADI scheme for a class of nonlinear parabolic isotropic systems. We establish an error estimate for this scheme, demonstrating its effectiveness in solving phase-field models. Our analysis provides rigorous mathematical justification for the numerical method’s reliability in simulating phase transitions. Full article

Tempered fractional diffusion equations constitute a critical class of partial differential equations with broad applications across multiple physical domains. In this paper, the Crank–Nicolson method and the tempered weighted and shifted Grünwald formula are used to discretize the tempered fractional diffusion equations. The discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite (SPD) Toeplitz matrix. For the discretized system, we propose a structure approximation-based preconditioning method. The structure approximation lies in two aspects: the inverse approximation based on the row-by-row strategy and the SPD Toeplitz approximation by the $\tau$ matrix. The proposed preconditioning method can be efficiently implemented using the discrete sine transform (DST). In spectral analysis, it is found that the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Numerical experiments demonstrate the effectiveness of the proposed preconditioner. Full article

A novel two–level linearized conservative finite difference method is proposed for solving the initial boundary value problem of the Rosenau–RLW equation. To preserve the energy conservation property, the Crank–Nicolson scheme is employed for temporal discretization, combined with an averaging treatment of the nonlinear term between the nth and $(n+1)$th time levels. For spatial discretization, a centered symmetric scheme is adopted. Meanwhile, the discrete conservation law is presented, and the existence and uniqueness of the numerical solutions are rigorously proved. Furthermore, the convergence and stability of the scheme are analyzed using the discrete energy method. Numerical experiments validate the theoretical results. Full article

This paper introduces a highly accurate and efficient conservative scheme for solving the nonlocal damped Schrödinger system with Riesz fractional derivatives. The proposed approach combines the Fourier spectral method with the Crank–Nicolson time-stepping scheme. To begin, the original equation is reformulated into an equivalent system by introducing a new variable that modifies both energy and mass. The Fourier spectral method is employed to achieve high spatial accuracy in this semi-discrete formulation. For time discretization, the Crank–Nicolson scheme is applied, ensuring conservation of the modified energy and mass in the fully discrete system. Numerical experiments validate the scheme’s precision and its ability to preserve conservation properties. Full article

The method of particular solutions using polynomial basis functions (MPS-PBF) has been extensively used to solve various types of partial differential equations. Traditional methods employing radial basis functions (RBFs)—such as Gaussian, multiquadric, and Matérn functions—often suffer from accuracy issues due to their dependence on a shape parameter, which is very difficult to select optimally. In this study, we adopt the MPS-PBF to solve the time-dependent Schrödinger equation in two dimensions. By utilizing polynomial basis functions, our approach eliminates the need to determine a shape parameter, thereby simplifying the solution process. Spatial discretization is performed using the MPS-PBF, while temporal discretization is handled via the backward Euler and Crank–Nicolson methods. To address the ill conditioning of the resulting system matrix, we incorporate a multi-scale technique. To validate the efficacy of the proposed scheme, we present four numerical examples and compare the results with known analytical solutions, demonstrating the accuracy and robustness of the scheme. Full article

This article describes an effective computing method for singularly perturbed parabolic problems with small negative shifts in convection and reaction terms. To handle the small negative shifts, the Taylor series expansion is used. The asymptotically equivalent singularly perturbed parabolic convection–diffusion–reaction problem is then discretized with the Crank–Nicolson method on a uniform mesh for the time derivative and a hybrid method on Shishkin-type meshes for the space derivative. The method’s stability and parameter-uniform convergence are established. To substantiate the theoretical findings, the numerical results are presented in tables and graphs are plotted. The present results improve the existing methods in the literature. Due to the effect of the small negative shifts in Examples 1 and 2, the numerical results using Shishkin and Bakhvalov–Shishkin meshes are almost the same. Since there are no small shifts in Examples 3 and 4, the numerical results using the Bakhvalov–Shishkin mesh are more efficient than using the Shishkin mesh. We conclude that the present method using the Bakhvalov–Shishkin mesh performs well for singularly perturbed problems without small negative shifts. Full article

We conducted a comprehensive comparative study of numerical solvers for the generalized Korteweg–de Vries (gKdV) equation, focusing on classical Fourier-based Crank–Nicolson methods and physics-informed neural networks (PINNs). Our work benchmarks these approaches across nonlinear regimes—including the cubic case ($\nu =3$)—and diverse initial conditions such as solitons, smooth pulses, discontinuities, and noisy profiles. In addition to pure PINN and spectral models, we propose a novel hybrid PINN–spectral method incorporating a regularization term based on Fourier reference solutions, leading to improved accuracy and stability. Numerical experiments show that while spectral methods achieve superior efficiency in structured domains, PINNs provide flexible, mesh-free alternatives for data-driven and irregular setups. The hybrid model achieves lower relative $L^2$ error and better captures soliton interactions. Our results demonstrate the complementary strengths of spectral and machine learning methods for nonlinear dispersive PDEs. Full article

This study presents a novel fifth-order iterative method for solving nonlinear systems derived from a modified combination of Jarratt and Newton schemes, incorporating a frozen derivative of the Jacobian. The method is applied to approximate solutions of the nonlinear convection–diffusion equation. A MATLAB script function was developed to implement the approach in two stages: first, discretizing the equation using the Crank–Nicolson Method, and second, solving the resulting nonlinear systems using Newton’s iterative method enhanced by a three-step Jarratt variant. A comprehensive analysis of the results highlights the method’s convergence and accuracy, comparing the numerical solution with the exact solution derived from linear parabolic partial differential transformations. This innovative fifth-order method provides an efficient numerical solution to the nonlinear convection–diffusion equation, addressing the problem through a systematic methodology that combines discretization and nonlinear equation solving. The study underscores the importance of advanced numerical techniques in tackling complex problems in physics and mathematics. Full article

This paper presents a fast Crank–Nicolson L1 finite difference scheme for the two-dimensional time fractional mobile/immobile diffusion equation with weakly singular solution at the initial moment. First, the time fractional derivative is discretized using the Crank–Nicolson formula on uniform meshes, and a local truncation error estimate is provided. The spatial derivative is discretized using the central difference quotient on uniform meshes. Then, energy analysis methods are utilized to provide an optimal error estimates. On the other hand, the numerical scheme is optimized based on the sum-of-exponentials approximation, effectively reducing computation and memory requirements. Finally, numerical examples are simulated to verify the effectiveness of the algorithm. Full article

The durability degradation of concrete structures in marine and urban underground environments is largely governed by chloride-induced corrosion. This process becomes significantly more severe under the coupled action of external loading and drying–wetting cycles, which accelerate chloride transport and structural deterioration. However, the existing research often isolates the effects of mechanical loading or environmental exposure, failing to comprehensively capture the synergistic interaction between these factors. This lack of understanding of chloride ingress under simultaneous mechanical and environmental loading limits the development of reliable service life prediction models for concrete structures. In this study, a self-made loading system was employed to simulate this coupled environment, combining external loading with 108 days of drying–wetting cycles. Chloride profiles were obtained to assess the combined effects of stress level, water/binder ratio, and fiber content on chloride penetration in fiber-reinforced cementitious composites (FRCCs). To further extend the analysis, a Crank–Nicolson-based finite difference approach was developed for the numerical assessment of chloride diffusion in concrete structures after repair. This model enables the point-wise treatment of nonlinear chloride concentration profiles and provides space- and time-dependent chloride concentration distributions. The results show that using an FRCC as a repair material significantly enhances the service life of chloride-contaminated concrete structures. The remaining service life of the repaired concrete was extended by 36.82% compared to the unrepaired case, demonstrating the clear practical value of FRCC repairs in aggressive environments. Full article

This paper is dedicated to investigating a highly accurate numerical solution for a class of 2D nonlinear time-dependent partial integro-differential equations with multi-term fractional integral items. These integrals are weakly singular with respect to time, which are handled using the product integration rule on graded meshes to compensate for the influence generated by the initial weak singular nature of the exact solution. The temporal derivative is approximated by a generalized Crank–Nicolson difference scheme, while the nonlinear term is approximated by a linearized method. Furthermore, the stability and convergence of the derived time semi-discretization scheme are strictly proved by revising the finite discrete parameters. Meanwhile, the differential matrices of the spatial high-order derivatives based on barycentric rational interpolation are utilized to obtain the fully discrete scheme. Finally, the effectiveness and reliability of the proposed method are validated by means of several numerical experiments. Full article

We study a system of ultra-cold dipolar Bose gas atoms confined in a two-dimensional (2D) harmonic trap with a dipolar impurity implanted at the center of the trap. Due to recent experimental progress in dipolar condensates, we focused on calculating properties of dipolar impurity systems that might guide experimentalists if they choose to study impurities in dipolar gases. We used the Gross–Pitaevskii formalism solved numerically via the split-step Crank–Nicolson method. We chose parameters of the background gas to be consistent with dysprosium (Dy), one of the strongest magnetic dipoles and of current experimental interest, and used chromium (Cr), erbium (Er), terbium (Tb), and Dy for the impurity. The dipole moments were aligned by an external field along what was chosen to be the z-axis, and we studied 2D confinements that were perpendicular or parallel to the external field. We show density contour plots for the two confinements, 1D cross-sections of the densities, calculated self-energies of the impurities while varying both number of atoms in the condensate and the symmetry of the trap. We also calculated the time evolution of the density of an initially pure system where an impurity is introduced. Our results show that while the self-energy increases in magnitude with increasing number of particles, it is reduced when the trap anisotropy follows the natural anisotropy of the gas, i.e., elongated along the z-axis in the case of parallel confinement. This work builds upon work conducted in Bose gases with zero-range interactions and demonstrates some of the features that could be found when exploring dipolar impurities in 2D Bose gases. Full article

In this paper, we discuss a class of nonlocal parabolic systems with nonlinear boundary conditions arising from the thermal explosion theory. First, we prove the local existence and uniqueness of the classical solution using the Leray–Schauder fixed-point theorem. Then, we analyze three Galerkin approximations of the system and derive the optimal-order error estimates: $\mathcal{O}(h^{r+1})$ in $L^2$ norm for continuous-time Galerkin approximation, $\mathcal{O}(h^{r+1}+{(\Delta t)}^2)$ in the $L^2$ norm for Crank–Nicolson Galerkin approximation, and $\mathcal{O}(h^{r+1}+{(\Delta t)}^2)$ in both $L^2$ and $H^1$ norms for extrapolated Crank–Nicolson Galerkin approximation. Full article

This research proposes a method for reducing the dimension of the coefficient vector for Crank–Nicolson mixed finite element (CNMFE) solutions to solve the fourth-order variable coefficient parabolic equation. Initially, the CNMFE schemes and corresponding matrix schemes for the equation are established, followed by a thorough discussion of the uniqueness, stability, and error estimates for the CNMFE solutions. Next, a matrix-form reduced-dimension CNMFE (RDCNMFE) method is developed utilizing proper orthogonal decomposition (POD) technology, with an in-depth discussion of the uniqueness, stability, and error estimates of the RDCNMFE solutions. The reduced-dimension method employs identical basis functions, unlike standard CNMFE methods. It significantly reduces the number of unknowns in the computations, thereby effectively decreasing computational time, while there is no loss of accuracy. Finally, numerical experiments are performed for both fourth-order and time-fractional fourth-order parabolic equations. The proposed method demonstrates its effectiveness not only for the fourth-order parabolic equations but also for time-fractional fourth-order parabolic equations, which further validate the universal applicability of the POD-based RDCNMFE method. Under a spatial discretization grid $40\times 40$, the traditional CNMFE method requires $2\times {41}^2$ degrees of freedom at each time step, while the RDCNMFE method reduces the degrees of freedom to $2\times 6$ through POD technology. The numerical results show that the RDCNMFE method is nearly 10 times faster than the traditional method. This clearly demonstrates the significant advantage of the RDCNMFE method in saving computational resources. Full article