Fourier Methods with Extended Stability Intervals for the Korteweg–de Vries Equation

Abstract

A full leap frog Fourier method for integrating the Korteweg–de Vries (KdV) equation $u_t + uu_x - \varepsilon u_{xxx} = 0$ results in an $O(N^{ - 3} )$ stability constraint on the time step, where N is the number of Fourier modes used. This stability limit is much more restrictive than the accuracy limit for many applications. In this paper we propose a method for which the stability limit is extended by treating the linear dispersive $u_{xxx} $ term implicitly. Thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit. The implicit method is implemented without solving linear systems by integrating in time in the Fourier space and discretizing the nonlinear $uu_x $ term by leap frog. A second method we propose uses basis functions which solve the linear part of the KdV equation and leap frog for time integration. A linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that the second scheme has an $O(N^{ - 1} )$ stability limit. The accuracy of the schemes for soliton propagation is analyzed by examining the truncation error. In addition, we analyze a linearization of a nonlinear finite element scheme proposed by Winther that treats the $u_{xxx} $ term by Crank–Nicolson. Numerical experiments on soliton solutions show that the linearized stability analysis gives accurate predictions for all the nonlinear schemes, that the truncation errors give an estimate for the accuracies and that the Fourier methods are more accurate than the finite element method.