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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.
Mots-clés : conservation laws, numerical methods, finite difference methods, central schemes
@article{M2AN_2005__39_5_965_0,
author = {Breuss, Michael},
title = {An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {965--993},
publisher = {EDP-Sciences},
volume = {39},
number = {5},
year = {2005},
doi = {10.1051/m2an:2005042},
zbl = {1077.35089},
mrnumber = {2178569},
language = {en},
url = {https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2005042/}
}
TY - JOUR AU - Breuss, Michael TI - An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2005 SP - 965 EP - 993 VL - 39 IS - 5 PB - EDP-Sciences UR - https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2005042/ DO - 10.1051/m2an:2005042 LA - en ID - M2AN_2005__39_5_965_0 ER -
%0 Journal Article %A Breuss, Michael %T An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2005 %P 965-993 %V 39 %N 5 %I EDP-Sciences %U https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2005042/ %R 10.1051/m2an:2005042 %G en %F M2AN_2005__39_5_965_0
Breuss, Michael. An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 39 (2005) no. 5, pp. 965-993. doi : 10.1051/m2an:2005042. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2005042/
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