The main goal of this paper is to develop an improved stochastic $\theta$-scheme as a numerical method for stochastic Volterra integral equations (SVIEs) with double weakly singular kernels and demonstrate that the stability of the proposed scheme is affected by the kernel parameters. To overcome the low computational efficiency of the stochastic $\theta$-scheme, we employed the sum-of-exponentials (SOE) approximation. Then, the mean square stability of the proposed scheme with respect to a convolution test equation is studied. Additionally, based on the stability conditions and the explicit structure of the stability matrices, analytical and numerical stability regions are plotted and compared with the split-step $\theta$-method and the $\theta$-Milstein method. The results confirm that our approach aligns significantly with the expected physical interpretations.

In this paper, we introduce a class of Runge-Kutta (RK) methods for backward stochastic differential equations (BSDEs). The convergence rate is studied and the corresponding order conditions are obtained. For the conditional expectations involved in the methods, we design an approximation algorithm by combining the characteristics of the methods and replacing the increments of Brownian motion with appropriate discrete random variables. An important feature of our approximation algorithm is that interpolation operations can be avoided. The numerical results of four examples are presented to show that our RK methods provide a good approach for solving the BSDEs.

We propose a finite difference approach to numerically solve an interface heat equation in one dimension with discontinuous conductivity and nonlinear interface condition. The discontinuous physical solution is sought among the multiple solutions of the nonlinear equation. Our method finds the approximate jump of the exact solution by two auxiliary linear problems with finite jumps. The approximate physical solution is then obtained by a weighted sum. The convergence and stability of the method are analyzed by the method of nonnegative matrices. Numerical examples are given to confirm the theory. In particular, numerical simulations are demonstrated in regards to the study of polymetric ion-selective electrodes and ion sensors.

In this article, we derive an optimal first-order Taylor-like formula. In a seminal paper [15], we introduced a new first-order Taylor-like formula that yields a reduced remainder compared to the classical Taylor’s formula. In this work, we relax the assumption of equally spaced points in our formula. Instead, we consider a sequence of unknown points and a sequence of unknown weights. We then solve an optimization problem to determine the optimal distribution of points and weights that minimizes the corresponding remainder. Numerical results are provided to illustrate our findings.

In this paper, we give a rigorous analysis of the finite difference time domain (FDTD) method with high accuracy in time (HAIT) (named HAIT-FDTD(M)) for the three dimensional Maxwell equations, where the time discretization is based on the Taylor expansion of the form: $U^n=C^0_n + C^1_n ∆t + · · · + \frac{1}{ M!}C^M_n (∆t)^M$ to approximate the fields in time. It is proven that the solutions of the schemes and the vectors representing the coefficients are divergence free. By using the energy method, the numerical energy identities of HAIT-FDTD(M) with $3 ≤ M ≤ 8$ are derived. It is then proved that these schemes are numerically and monotonically energy conserved as the polynomial degree $M$ becomes large. With the help of the energy identities, stability conditions for the six schemes are derived, and how to select $M$ and $∆t$ in practice is given. By deriving error equations, we prove that the six schemes have convergence of the $M{\rm th}$ order in time and the second order in space. Numerical experiments are provided and confirm the analysis on free divergence, approximate energy conservation, stability, and convergence.

The paper introduces a fourth-order augmented matched interface and boundary (AMIB) method for solving elliptic interface problems with complex interfaces and piecewise smooth coefficients in two and three dimensions. To resolve the challenge posed by non-constant coefficients within the AMIB framework, the fast Fourier transform (FFT) Poisson solver of the existing AMIB methods is replaced by a geometric multigrid method to efficiently invert the Laplacian discretization matrix. In this work, a fourth order multigrid method will be employed in the framework of the AMIB method for elliptic interface problems with variable coefficients in two and three dimensions. Based on a Cartesian mesh, the standard fourth-order finite differences are employed to approximate the first and second derivatives involved in the Laplacian with variable coefficients. Near the interface, a fourth-order ray-casting matched interface and boundary (MIB) scheme is generalized to variable coefficient problems to enforce interface jump conditions in the corrected finite difference discretization. The augmented formulation of the AMIB allows us to decouple the interface treatments from the inversion of the Laplacian discretization matrix, so that one essentially solves an elliptic subproblem without interfaces. A fourth order geometric multigrid method is introduced to solve this subproblem with a Dirichlet boundary condition, where fourth order one-sided finite difference approximations are considered near the boundary in all grid levels. The proposed multigrid method significantly enhances the computational efficiency in solving variable coefficient problems, while achieving a fourth-order accuracy in accommodating complex interfaces and discontinuous solutions.

The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using an $m$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the source term is nonsmooth in time and incompatible with the initial data. Numerical results are presented to support the theoretical results. The proposed approach is applicable for stochastic subdiffusion equation.