Solitary waves of the Korteweg-de Vries-Burgers' equation
type: Item
instance of: scholarly article
title: Solitary waves of the Korteweg-de Vries-Burgers' equation
zbMATH Open document ID: 0951.65097
DOI: 10.1016/S0010-4655(99)00470-1
author: S. I. Zaki
published in: Computer Physics Communications
publication date: 2000-06-21T00:00:00Z
review text: The author studies the numerical solution of the singularly perturbed Korteweg-de Vries-Burgers equation in one spatial dimension, \[ u_t-\nu u_{xx}+\mu u_{xxx}+\varepsilon uu_x= 0,\quad x\in [a,b],\quad t>0,\quad \nu,\mu,\varepsilon> 0. \] For the spatial discretization, a finite element method using cubic B-splines as ansatz and test functions is proposed. Appearing integrals are approximated by means of a Gaussian quadrature formula. The discretization in time is a Crank-Nicolson scheme. The nonlinearity is resolved by an inner iteration. The method is shown to be unconditionally stable in the sense of von Neumann. Numerical tests are presented for the cases \(\nu= 0\) (Korteweg-de Vries equation), \(\mu=0\) and small \(\nu\) (Burgers equation), and \(\nu,\mu\neq 0\) (Korteweg-de Vries-Burgers equation). The test for the Burgers equation shows high accuracy. The numerical solution of the Korteweg-de Vries test equation also fulfills conservation principles known for the exact solution.
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 76B25
zbMATH DE Number: 1462870
zbMATH Keywords: solitary waves
zbMATH Keywords: von Neumann stability
zbMATH Keywords: numerical examples
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: Burgers equation
zbMATH Keywords: Korteweg-de Vries-Burgers equation
zbMATH Keywords: finite element method
zbMATH Keywords: cubic B-splines
zbMATH Keywords: Gaussian quadrature formula
zbMATH Keywords: Crank-Nicolson scheme
zbMATH Keywords: inner iteration
reviewed by: Etienne Emmrich
MaRDI profile type: Publication
full work available at URL: s0010-4655(99)00470-1
OpenAlex ID: W1985245383
description: scientific article; zbMATH DE number 1462870