Parametric excitation of computational modes inherent to leap-frog scheme applied to the Korteweg-de Vries equation
The problem studied in this paper is the numerical solution of the Korteweg-de Vries equation \(\partial u/\partial t+u(\partial u/\partial x)+\mu (\partial^ 3u/\partial x^ 3)=0\) with periodic boundary conditions \(u(x,t)=u(x+2,t),\) and initial conditions \(u(x,0)=2\epsilon_ k\cos (\pi kx).\) The investigated numerical method is the so called leap- frog scheme: \(u_ j^{n+1}=u_ j^{n-1}-(\Delta t/3\Delta x)(u^ n_{j+1}+u^ n_ j+u^ n_{j-1})(u^ n_{j+1}-u^ n_{j-1})-\mu (\Delta t/\Delta x^ 3)(u^ n_{j+2}-2u^ n_{j+1}+2u^ n_{j-1}- u^ n_{j-2}).\) It is shown that the numerical solution obtained by this method consists of physical and computational components. Moreover the authors can to separate these components. A method to suppress the computational component is given using a more exact explicit one-step method combined with some implicit step to determine \(u^{n+1}\). The effectiveness of this method is demonstrated by several computational results.
scientific article; zbMATH DE number 4119571