2014•Computational methods for differential equationsOpen access

A new total variation diminishing implicit nonstandard finite difference scheme for conservation laws

Mohammad Mehdizadeh Khalsaraei, Fayyaz Khodadosti

Abstract

In this paper, a new implicit nonstandard finite difference scheme for conservation laws, which preserving the property of TVD (total variation diminishing) of the solution, is proposed. This scheme is derived by using nonlocal approximation for nonlinear terms of partial differential equation. Schemes preserving the essential physical property of TVD are of great importance in practice. Such schemes are free of spurious oscillations around discontinuities. Numerical results for Burger's equation is presented. Comparison of numerical results with a classical difference scheme is given.

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Key concepts: Conservation law, Total variation diminishing, Mathematics, Classification of discontinuities, Partial differential equation, Finite difference, Nonlinear system, Scheme (mathematics)