Numerical solution of two-dimensional sine-Gordon and MBE models using Fourier spectral and high order explicit time stepping methods

Abstract

In this paper, we consider a Fourier spectral method for spatial variables and three fast and efficient explicit time stepping methods for the numerical solution of two-dimensional (2D) nonlinear partial differential equations (PDEs) including sine-Gordon and molecular beam epitaxy (MBE) models. We discretize the original PDEs with discrete Fourier transform in space and then apply some fourth-order time stepping schemes such as Runge–Kutta integrating factor (IFRK), Runge–Kutta exponential time differencing (ETDRK) and ETDRK method with improved accuracy (ETDRKB), to the resulting systems of ordinary differential equations (ODEs). With regard to stiffness and nonlinearities of the models, in this approach we take advantage of using the fast Fourier transform in combination with explicit fourth-order time stepping methods and expect to get a better accuracy and CPU time in comparison with other reported spatial discretizations and implicit time stepping schemes in literature. Comparing the numerical solutions with analytical solutions and also some reported results in the literature demonstrates the effectiveness of the proposed schemes. We will also show that the schemes of this paper exhibit the conservation of energy for both PDEs.