Total variation diminishing nonstandard finite difference schemes for conservation laws

Authors:
Roumen Anguelov;Jean M. -S. Lubuma;Froduald Minani
Affiliations:
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa;Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa;Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
Venue:
Mathematical and Computer Modelling: An International Journal
Year:
2010

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Abstract

Nonstandard finite difference schemes for conservation laws preserving the property of diminishing total variation of the solution are proposed. Computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step-size.