New Time Differencing Methods for Spectral Methods

Abstract

A new semi-analytical time differencing is applied to spectral methods for partial differential equations which involve higher spatial derivatives. This is developed in Jung and Nguyen (J Sci Comput (2015) 63:355-373) based on the classical integrating factor (IF) and exponential time differencing (ETD) methods. The basic idea is approximating analytically the stiffness (fast part) by the so-called correctors (see 1.3 below) and numerically the non-stiffness (slow part) by the IF and ETD, etc. It turns out that rapid decay and rapid oscillatory modes in the spectral methods are well approximated by our corrector methods, which in turn provides better accuracy in the numerical schemes presented in the text. We investigate some nonlinear problems with a quadratic nonlinear term, which makes all Fourier modes interact with each other. We construct the correctors recursively to accurately capture the stiffness in the mode interactions. Polynomial or other types of nonlinear interactions can be tackled in a similar fashion.