Breuss, Michael An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws. (English) Zbl 1077.35089 ESAIM, Math. Model. Numer. Anal. 39, No. 5, 965-993 (2005). Summary: We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov’s method and the staggered and non-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema. The dependence of oscillatory properties on the numerical viscosity coefficient is investigated rigorously for the LFt schemes, illuminating also the properties of Rusanov’s method. It turns out, that schemes with a large viscosity coefficient are prone to oscillations at data extrema. For all LFt schemes except for the classical Lax-Friedrichs method, occurring oscillations are damped in the course of a computation. This damping effect also holds for Rusanov’s method. Concerning the NT schemes, the non-staggered version may yield oscillatory results, while it can be shown rigorously that the staggered NT scheme does not produce oscillations when using the classical minmod-limiter under a restriction on the time step size. Note that this restriction is not the same as the condition ensuring the TVD property. Numerical investigations of one-dimensional scalar problems and of the system of shallow water equations in two dimensions with respect to the phenomenon complete the paper. Reviewer: Qin Mengzhao (Beijing) Cited in 6 Documents MSC: 35L65 Hyperbolic conservation laws 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs Keywords:Lax-Friedriches scheme; Rusanov’s scheme; Nessyahu-Tadmor scheme; minmod-limiter; shallow water equation PDFBibTeX XMLCite \textit{M. Breuss}, ESAIM, Math. Model. Numer. Anal. 39, No. 5, 965--993 (2005; Zbl 1077.35089) Full Text: DOI Numdam EuDML References: [1] M. Breuß , The correct use of the Lax-Friedrichs method . ESAIM: M2AN 38 ( 2004 ) 519 - 540 . Numdam | Zbl 1077.65089 · Zbl 1077.65089 [2] L. Evans , Partial Differential Equations . American Mathematical Society ( 1998 ). MR 1625845 | Zbl 0902.35002 · Zbl 0902.35002 [3] E. Godlewski and P.-A. Raviart , Hyperbolic systems of conservation laws . Edition Marketing ( 1991 ). 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