An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws

Michael Breuss

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 5, page 965-993
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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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 39.5 (2010): 965-993. <http://eudml.org/doc/194295>.

@article{Breuss2010,
abstract = { 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. },
author = {Breuss, Michael},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Conservation laws; numerical methods; finite difference methods; central schemes.; Lax-Friedriches scheme; Rusanov's scheme; Nessyahu-Tadmor scheme; minmod-limiter; shallow water equation},
language = {eng},
month = {3},
number = {5},
pages = {965-993},
publisher = {EDP Sciences},
title = {An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws},
url = {http://eudml.org/doc/194295},
volume = {39},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 5
SP - 965
EP - 993
AB - 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.
LA - eng
KW - Conservation laws; numerical methods; finite difference methods; central schemes.; Lax-Friedriches scheme; Rusanov's scheme; Nessyahu-Tadmor scheme; minmod-limiter; shallow water equation
UR - http://eudml.org/doc/194295
ER -

References

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  13. D. Levy, G. Puppo and G. Russo, Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput.22 (2000) 656–672.  
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