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 Driscolls 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 that for these applications with fixed time steps, the modified ETD scheme is the best.