Fourth-Order Time-Stepping for Stiff PDEs

A modification of the ETDRK4 (Exponential Time Di#erencing fourth-order RungeKutta) method for solving sti# nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified ETD scheme against the competing methods of implicit--explicit di#erencing, integrating factors, time-splitting, and Fornberg and Driscoll's "sliders" for the KdV, Kuramoto-Sivashinsky, Burgers, and Allen-Cahn equations in one space dimension. Implementation of the method is illustrated by short Matlab programs for two of the equations. It is found…