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A Sylvester-Based IMEX Method via Differentiation Matrices for Solving Nonlinear Parabolic Equations

Published online by Cambridge University Press:  03 June 2015

Francisco de la Hoz  and
Fernando Vadillo
Francisco de la Hoz*
Affiliation:
Department of Applied Mathematics, Statistics and Operations Research, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940 Leioa, Spain
Fernando Vadillo*
Affiliation:
Department of Applied Mathematics, Statistics and Operations Research, Faculty of Science and Technology, University of the Basque Country UPV/EHU, Barrio Sarriena S/N, 48940 Leioa, Spain
*
Corresponding author.Email:francisco.delahoz@ehu.es
Email:fernando.vadillo@ehu.es
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Abstract

In this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains, taking Hermite functions, sinc functions, and rational Chebyshev polynomials as basis functions. The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme, being of particular interest the treatment of three-dimensional Sylvester equations that we make. The resulting method is easy to understand and express, and can be implemented in a transparent way by means of a few lines of code. We test numerically the three choices of basis functions, showing the convenience of this new approach, especially when rational Chebyshev polynomials are considered.

Keywords

65M2065M7002.70.Hm02.30.JrSemi-linear diffusion equationspseudo-spectral methodsdifferentiation matricesHermite functionssinc functionsrational Chebyshev polynomialsIMEX methodsSylvester equationsblow-up

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 4 , October 2013 , pp. 1001 - 1026
Copyright
Copyright © Global Science Press Limited 2013

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