Stability Analysis and Application of the Exponential Time Differencing Schemes
Qiang Du 1, Wen Xiang Zhu 21 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA/ LSEC, ICMSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China
2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Exponential time differencing schemes are time integration methods that can be effi- ciently combined with spatial spectral approximations to provide very high resolution to the smooth solutions of some linear and nonlinear partial differential equations. We study in this paper the stability properties of some exponential time differencing schemes. We also present their application to the numerical solution of the scalar Allen-Cahn equation in two and three dimensional spaces.
Key words: Time integration schemes; Exponential time differencing; Fourier spectral methods; Stability; Fourier analysis; Energy estimates; Maximum principle; Allen-Cahn equations; Phase transitions.