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Convenient total variation diminishing conditions for nonlinear difference schemes. (English) Zbl 0662.65082
Scalar conservation laws \(u(x,t)_ t+[f(u(x,t))]_ x=0\) are considered and discrete difference approximations are proposed and studied which produce approximate solutions having the property that their total variation diminishes. Thus alternative characterizations of such difference schemes are provided, and it is shown that the total variation of a grid function depends solely on its extreme values. Hence the sufficient conditions must be satisfied only in the neighborhoods of extreme values, and the mentioned property holds for difference schemes whose numerical viscosity corresponds to upwind differcing at extreme values but can be arbitrary otherwise. In the paper semidiscrete and fully discrete implicit and explicit schemes are discussed.
Reviewer: H.R.Schwarz

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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