Parametric excitation of computational modes inherent to leap-frog scheme applied to the Korteweg-de Vries equation
type: Item
instance of: scholarly article
title: Parametric excitation of computational modes inherent to leap-frog scheme applied to the Korteweg-de Vries equation
zbMATH Open document ID: 0683.65104
DOI: 10.1016/0021-9991(89)90130-7
author: Akira Aoyagi
author: Kanji Abe
published in: Journal of Computational Physics
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4119571
zbMATH Keywords: nonlinear instability
zbMATH Keywords: finite difference schemes
zbMATH Keywords: numerical examples
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: leap-frog scheme
zbMATH Keywords: explicit one-step method
MaRDI profile type: Publication
full work available at URL: 0021-9991(89)90130-7
OpenAlex ID: W2011504711
cites work: The Stability Properties of a Coupled Pair of Non-Linear Partial Difference Equations
cites work: Heuristic stability theory for finite-difference equations
cites work: On the Instability of Leap-Frog and Crank-Nicolson Approximations of a Nonlinear Partial Differential Equation
cites work: Studies in numerical nonlinear instability. II. A new look at \(u_ t+uu_ x=0\)
cites work: On nonlinear instabilities in leap-frog finite difference schemes
cites work: Focusing: A mechanism for instability of nonlinear finite difference equations
cites work: Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
cites work: Method for Solving the Korteweg-deVries Equation
description: scientific article; zbMATH DE number 4119571