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Total-variation-diminishing time discretizations. (English) Zbl 0662.65081
For the approximate solution of hyperbolic conservation laws \(u_ t+\sum^{d}_{i=1}f_ i(u)_{x_ i}=0\) for \(u\in {\mathbb{R}}^ m\) and \(x\in {\mathbb{R}}^ d\) difference schemes with the property of diminishing the total variation are a successful tool. In the present paper the scalar, one-dimensional case is considered and discussed, only. For steady state calculations, in which time accuracy is not important, a class of Runge-Kutta m-step time discretizations with a large CFL number is presented. On the other hand, for time-dependent problems a class of high order multilevel time discretizations with the total variation diminishing property is outlined.
Reviewer: H.R.Schwarz

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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