相关论文: Fully discrete finite difference schemes for the F…

In this paper, we study the stability and convergence of a fully discrete finite difference scheme for the initial value problem associated with the Korteweg-De Vries (KdV) equation. We employ the Crank-Nicolson method for temporal…

In this paper we study the convergence of a fully discrete Crank-Nicolson Galerkin scheme for the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and non-linear…

We present a fully discrete Crank-Nicolson Fourier-spectral-Galerkin (FSG) scheme for approximating solutions of the fractional Korteweg-de Vries (KdV) equation, which involves a fractional Laplacian with exponent $\alpha \in [1,2]$ and a…

We propose a local discontinuous Galerkin (LDG) method for the fractional Korteweg-de Vries (KdV) equation, involving the fractional Laplacian with exponent $\alpha \in (1,2)$ in one and multiple space dimensions. By decomposing the…

Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic…

We prove convergence of a fully discrete finite difference scheme for the Korteweg--de Vries equation. Both the decaying case on the full line and the periodic case are considered. If the initial data $u|_{t=0}=u_0$ is of high regularity,…

The Korteweg-de Vries (KdV) equation is a fundamental partial differential equation that models wave propagation in shallow water and other dispersive media. Accurately solving the KdV equation is essential for understanding wave dynamics…

We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the…

This article presents a finite element scheme with Newton's method for solving the time-fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank-Nicolson scheme based on backward Euler convolution…

In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic L\'evy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of…

In this paper, we introduce a conservative Crank-Nicolson-type finite difference schemes for the regularized logarithmic Schr\"{o}dinger equation (RLSE) with Dirac delta potential in 1D. The regularized logarithmic Schr\"{o}dinger equation…

In this paper, we study loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference scheme of Crank-Nicholson type of higher order is developed for…

We apply a composite idea of semi-discrete finite difference approximation in time and Galerkin finite element method in space to solve the Navier-Stokes equations with Caputo derivative of order 0 < {\alpha} < 1. The stability properties…

In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed…

Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of…

Numerical schemes that conserve invariants have demonstrated superior performance in various contexts, and several unified methods have been developed for constructing such schemes. However, the mathematical properties of these schemes…

This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential…

In this article we propose two finite element schemes for the Navier-Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak…

This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional…

For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for…