On an exact numerical simulation of solitary-wave solutions of the Burgers-Huxley equation through Cardano's method
type: Item
instance of: scholarly article
title: On an exact numerical simulation of solitary-wave solutions of the Burgers-Huxley equation through Cardano's method
zbMATH Open document ID: 1306.65244
published in: BIT
publication date: 2014-10-31T00:00:00Z
review text: This article deals with one-dimensional Burgers-Huxley equations where two analytical solutions are known. The determination of exact solitary wave solutions remains nevertheless an open problem. The author proposes an exact numerical approximation which provides an adequate model for analytical solutions and therefore also for proposed solitary wave solutions. The approach is shortly described as following: the author discretizes the equation by a symmetric finite difference scheme in spatial and temporal directions. He computes then the approximate solution for the next temporal node by exactly solving a cubic polynomial equation arising from the spatial discretization. The author proves that the solutions, derived by this method under suitable constraints on the arameters, has the same mathematical properties as the analytical solutions: they are positive, bounded and monotone in spatial and temporal direction. By simulation results he confirms further that his approach provides a good approximations of the analytical solutions. His method, by using a symmetric finite difference scheme, is limited to problems in which advection is not dominating. For advection-dominated problems one should use instead upwind or downwind discretization schemes to avoid instabilities.
reviewed by: Dimitris P. Vartziotis
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35Q53
zbMATH DE Number: 6363087
zbMATH Keywords: Burgers-Huxley equation
zbMATH Keywords: nonlinear finite difference method
zbMATH Keywords: solitary-wave solutions
zbMATH Keywords: Cardano's formulas
zbMATH Keywords: positivity
zbMATH Keywords: boundedness
zbMATH Keywords: monotonicity
MaRDI profile type: Publication
full work available at URL: s10543-013-0466-9
OpenAlex ID: W1972144460
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DOI: 10.1007/S10543-013-0466-9
author: J. E. Macías-Díaz
description: scientific article; zbMATH DE number 6363087