Fourth Order Time-Stepping for Kadomtsev–Petviashvili and Davey–Stewartson Equations

Abstract

Purely dispersive partial differential equations such as the Korteweg–de Vries equation, the nonlinear Schrödinger equation, and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. To numerically study such phenomena, fourth order time-stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Kadomtsev–Petviashvili and the Davey–Stewartson equations, two integrable equations in $2+1$ dimensions: these methods are exponential time-differencing, integrating factors, time-splitting, implicit Runge–Kutta, and Driscoll's composite Runge–Kutta method. The accuracy in the numerical conservation of integrals of motion is discussed.