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A positivity‐preserving nonstandard finite difference scheme for the damped wave equation
Author(s) -
Mickens Ronald E.,
Jordan P. M.
Publication year - 2004
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20003
Subject(s) - mathematics , heat equation , mathematical analysis , partial differential equation , hyperbolic partial differential equation , boundary value problem , stability (learning theory) , damped wave , wave equation , nonlinear system , finite difference , finite difference method , thermal conduction , initial value problem , ftcs scheme , fourier transform , differential equation , physics , differential algebraic equation , ordinary differential equation , quantum mechanics , machine learning , computer science , thermodynamics
A positivity‐preserving nonstandard finite difference scheme is constructed to solve an initial‐boundary value problem involving heat transfer described by the Maxwell‐Cattaneo thermal conduction law, i.e., a modified form of the classical Fourier flux relation. The resulting heat transport equation is the damped wave equation, a PDE of hyperbolic type. In addition, exact analytical solutions are given, special cases are mentioned, and it is noted that the positivity condition is equivalent to the usual linear stability criteria. Finally, solution profiles are plotted and possible extensions to a delayed diffusion equation and nonlinear reaction‐diffusion systems are discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.