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Contributions to the mathematics of the nonstandard finite difference method and applications
Author(s) -
Anguelov Roumen,
Lubuma Jean M.S.
Publication year - 2001
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.1025
Subject(s) - mathematics , partial derivative , stability (learning theory) , finite difference , partial differential equation , scheme (mathematics) , nonlinear system , order (exchange) , differential equation , finite difference method , mathematical analysis , computer science , physics , finance , quantum mechanics , machine learning , economics
We formalize the transfer of essential properties of the solution of a differential equation to the solution of a discrete scheme as qualitative stability with respect to the properties. This permits us to motivate some rules (viz. on the order of the difference equation, on the renormalization of the denominator of the discrete derivative, and on nonlocal approximation of nonlinear terms) used in the design of nonstandard finite difference schemes. Extensions of some models are considered, and numerical examples confirming the efficiency of the nonstandard approach are provided. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 518–543, 2001