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

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Abstract
A full leap frog Fourier method for integrating the Korteweg-de Vries (KdV) equation u//t plus uu//x minus epsilon u//x//x//x equals 0 results in an O(N** minus **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. The authors propose a method for which the staibility limit is extended by treating the linear dispersive u//x//x//x 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 they propose uses basis functions which solve the linear part of the KdV equation and leap frog for time integration

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