# 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 (2010)

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

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topBreuss, 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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