Exponential integrator Fourier Galerkin methods for semilinear parabolic equations

TL;DR

This paper introduces the EIFG method, combining Fourier Galerkin spatial discretization with exponential Runge-Kutta time integration, to improve accuracy in solving semilinear parabolic equations.

Contribution

The paper proposes a novel exponential integrator Fourier Galerkin method that enhances spatial accuracy and provides explicit error estimates for semilinear parabolic equations.

Findings

Error estimates in $H^2$-norm are derived for the method.

Numerical examples confirm the method's high accuracy and efficiency.

The method performs well in both 2D and 3D problems.

Abstract

In this paper, in order to improve the spatial accuracy, the exponential integrator Fourier Galerkin method (EIFG) is proposed for solving semilinear parabolic equations in rectangular domains. In this proposed method, the spatial discretization is first carried out by the Fourier-based Galerkin approximation, and then the time integration of the resulting semi-discrete system is approximated by the explicit exponential Runge-Kutta approach, which leads to the fully-discrete numerical solution. With certain regularity assumptions on the model problem, error estimate measured in $H^2$-norm is explicitly derived for EIFG method with two RK stages. Several two and three dimensional examples are shown to demonstrate the excellent performance of EIFG method, which are coincident to the theoretical results.