A Numerical Study for Improving Time Step Methods in Psudospectral Schemes Applied to the Korteweg and De Vries Equation

Abstract

The numerical integration of partial differential equations that admit solitons like solutions appears frequently in science as the case of the equation of Korteweg and de Vries (KdV).
The nonlinear behavior and the existence of solitons and interacting solitons constitute an important tool for the study of the propagation of that kind of nonlinear traveling waves. On the other hand, the linear term is the one in which arise the main restriction for the time step setup.
The time expended in the computational running is important in particular when the time integration is done by an explicit method; although implicit schemes exist, the explicit ones are often used, because they are easy to codify in almost every computer language.
To reduce the time required for the numerical integration, one proposal found in the literature is known as the integrating factor (IF), another is based upon the method named exponential time differencing (ETD). The second one represents the state of the art in improving time stepping in explicit methods for non linear differential equations.
In the present work a pseudospectral method was implemented for the KdV equation, and with the aid of the IF and ETD a several numerical experiments was performed.
The numerical study includes an analysis of the global error depending of the time step selection and the variation of results among a few explicit numerical schemes of high order of precision for the time integration.