ABSTRACT This work presents a comparative study of efficient computational methods for time‐fractional singularly perturbed parabolic problems. The solutions to such problems are characterized by an initial layer near , arising from the weak singularity of the fractional temporal derivative, and a boundary layer induced by the presence of a small perturbation parameter in the spatial operator. To accurately capture these multiscale features, three numerical schemes are developed and analyzed on layer‐adapted meshes, highlighting the inadequacy of uniform meshes for this class of problems. In the L1‐UP scheme, the fractional operator is approximated using the classical L1 method on a graded temporal mesh, while the spatial layer is resolved using an upwind difference discretization on a piecewise‐uniform Shishkin mesh. The accuracy of this scheme is further enhanced via a post‐processing extrapolation technique, leading to the L1‐EXP scheme with improved convergence properties. To achieve higher spatial accuracy, a hybrid discretization combining midpoint upwind and central difference techniques is proposed in the L1‐HYB scheme, yielding second‐order accuracy in space. All proposed schemes are shown to be parameter‐uniform and robust with respect to the perturbation parameter. Their performance is further assessed using a Bakhvalov‐Shishkin mesh, which provides improved resolution within the boundary layer. Comprehensive numerical experiments are conducted to validate the theoretical convergence results and to illustrate the effectiveness of the derived methods in accurately resolving both initial and boundary layers.

ABSTRACT This work focuses on a posteriori error estimation within a non‐conforming framework for the stabilizer‐free immersed weak Galerkin finite element method (IWG‐FEM) applied to elliptic and parabolic problems with non‐smooth coefficients. With an appropriate adaptation of the Helmholtz decomposition of the error, an optimal a posteriori error bound is derived in the weighted ‐seminorm to address elliptic problems with non‐smooth coefficients. For parabolic problems, we analyze a fully discrete scheme that combines the implicit backward Euler time‐stepping method with the non‐conforming IWG‐FEM. A reliable a posteriori error indicator is derived using the time‐dependent Helmholtz decomposition in the ‐norm in time and the weighted ‐seminorm in space without directly relying on the energy argument. Finally, numerical experiments on various test problems confirm the convergence behavior of the proposed error indicators.

ABSTRACT By using the local Radial Basis Function‐based Differential Quadrature method (LRBF‐DQ), a numerical simulation technique for Gray–Scott Reaction‐Diffusion (RD) systems was proposed. This paper has used the LRBF‐DQ method for space discretization with Crank–Nicolson format combined with nonlinear processing for time discretization to obtain the numerical format. In addition, the stability of this format was also analyzed by plotting the eigenvalue of the iteration matrix, and an unconditionally stable result was obtained. The LRBF‐DQ method combined with nonlinear processing was used for the first time to simulate the 2D Gray–Scott RD system in this article. After numerical testing, it was found that under the same conditions, this method has higher accuracy and can simulate better spot images with fewer points and longer time intervals.

ABSTRACTThe Crank–Nicolson radial basis function (CN‐RBF) method, as a time‐stepping mesh‐less approach, is widely adopted for multi‐asset option pricing. However, its accuracy is highly dependent on shape parameters, which limits its practical applicability. To address this, we propose two polynomial‐based space‐time methods for pricing European exchange options: one is the space‐time polynomial particular solutions method, and the other is the space‐time interpolations method which directly uses polynomial functions as basis functions. These two methods treat the time variable as a common spatial variable, thereby eliminating the need for the time‐stepping process. Their implementation requires only interpolation points, simplifying the solution procedure by removing shape parameter selection and time‐step scheme dependencies. Numerical results demonstrate that both space‐time methods achieve good performance in terms of accuracy and stability.

ABSTRACT This paper proposes a discrete Laplace transform‐based L1 scheme for fractional time‐delay diffusion‐reaction equations with multiple delay‐induced singularities. A pointwise error analysis establishes that the convergence rate at is , where , is the delay parameter, and is the order of the Caputo derivative. Numerical experiments confirm theoretical predictions and demonstrate the effectiveness of the method in resolving singular behavior between delay interfaces.

ABSTRACTIn this article, we carry out a systematic stability analysis for the explicit Runge–Kutta scheme coupled with a discontinuous Galerkin discretization in space for solving the time‐dependent Maxwell's equations. Many so‐called RKDG methods have been developed and successfully solved Maxwell's equations. However, to our best knowledge, there are quite limited rigorous analyses of RKDG methods devoted to Maxwell's equations. This paper is our initial effort in filling this gap by establishing the stability analysis for the RKDG methods of order two, three, and four. Numerical results consistent with our analysis are presented for the 3‐D Maxwell's equations.

ABSTRACTIn the article, we study an efficient numerical scheme to solve a class of space fractional Klein‐Gordon‐Schrödinger equations with periodic boundary condition. First, we propose finite difference scheme to discrete time derivative, and the space fractional derivative is approximated by Fourier spectral collocation method. Second, we prove that the arising numerical scheme preserves discrete mass and energy conservation laws, respectively. Moreover, the existence, uniqueness, and convergence of the obtained numerical scheme are discussed, and it is shown that the scheme is of the accuracy . However, the resulting numerical scheme faces how to efficiently solve a large fully implicit system at each time step, and it takes too much time in the numerical simulation. Then, we introduce split‐step Fourier spectral collocation method to improve calculation speed. Finally, the numerical experiments including one‐dimensional and two‐dimensional fractional Klein‐Gordon‐Schrödinger systems are given to verify the correctness of theoretical results.

Numerical Methods for Partial Differential Equations is an international journal that publishes the highest quality research in the rigorous analysis of novel techniques for the numerical solution of partial differential equations (PDEs).The journal is intended to be accessible to a broad spectrum of researchers into numerical approximation of PDEs throughout science and engineering, with priority given to articles that focus on theoretical results describing robustness, stability, and convergence of the new methods rather than the techniques themselves and specific applications.The Journal seeks to be interdisciplinary, while emphasizing numerical analysis and approximation theory in the following areas of research: discretization schemes for linear, nonlinear, and fractional PDEs; methods for optimal control and parameter estimation problems; techniques for high-dimensional spatial and parameterized PDEs; learning algorithms for data-driven solutions to PDEs; and new approaches for modeling complex phenomena with PDEs.