Maximum norm error analysis of an unconditionally stable semi‐implicit scheme for multi‐dimensional Allen–Cahn equations

Abstract

In this paper, a linearized finite difference scheme is proposed for solving the multi‐dimensional Allen–Cahn equation. In the scheme, a modified leap‐frog scheme is used for the time discretization, the nonlinear term is treated in a semi‐implicit way, and the central difference scheme is used for the discretization in space. The proposed method satisfies the discrete energy decay property and is unconditionally stable. Moreover, a maximum norm error analysis is carried out in a rigorous way to show that the method is second‐order accurate both in time and space variables. Finally, numerical tests for both two‐ and three‐dimensional problems are provided to confirm our theoretical findings.

A fourth-order linearized difference scheme for the coupled space fractional Ginzburg–Landau equation

“…A three-level linearized difference scheme [26][27][28][29] for boundary value problem (1)- (3) with homogeneous Dirichlet boundary conditions (12) in the finite domain Ω = [a, b] is as follows:…”