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 - Modélisation Mathématique et Analyse Numérique (2005)

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

Abstract

top
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

top

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 - Modélisation Mathématique et Analyse Numérique 39.5 (2005): 965-993. <http://eudml.org/doc/245448>.

@article{Breuss2005,
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 - Modélisation Mathématique et Analyse Numérique},
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},
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/245448},
volume = {39},
year = {2005},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2005
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/245448
ER -

References

top
  1. [1] M. Breuß, The correct use of the Lax-Friedrichs method. ESAIM: M2AN 38 (2004) 519–540. Zbl1077.65089
  2. [2] L. Evans, Partial Differential Equations. American Mathematical Society (1998). Zbl0902.35002MR1625845
  3. [3] E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Edition Marketing (1991). Zbl0768.35059MR1304494
  4. [4] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer Verlag, New York (1996). Zbl0860.65075MR1410987
  5. [5] A. Harten, On a class of high order resolution total variation stable finite difference schemes. SIAM J. Numer. Anal. 21 (1984) 1–23. Zbl0547.65062
  6. [6] G.-S. Jiang and E. Tadmor, Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (1998) 1892–1917. Zbl0914.65095
  7. [7] G.-S. Jiang, D. Levy, C.T. Lin, S. Osher and E. Tadmor, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM J. Numer. Anal. 35 (1998) 2147–2168. Zbl0920.65053
  8. [8] S. Jin and Z. Xin, The relaxation scheme for systems of conservation laws in arbitrary space dimension. Comm. Pure Appl. Math. 45 (1995) 235–276. Zbl0826.65078
  9. [9] P.D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical approximation. Comm. Pure Appl. Math. 7 (1954) 159–193. Zbl0055.19404
  10. [10] P.G. Lefloch and J.-G. Liu, Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions. Math. Comp. 68 (1999) 1025–1055. Zbl0915.35069
  11. [11] R.J. Leveque, Numerical Methods for Conservation Laws. Birkhäuser Verlag, 2nd Edition (1992). Zbl0847.65053MR1153252
  12. [12] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). Zbl1010.65040MR1925043
  13. [13] D. Levy, G. Puppo and G. Russo, Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput. 22 (2000) 656–672. Zbl0967.65089
  14. [14] X.D. Liu and E. Tadmor, Third order nonoscillatory central schemes for hyperbolic conservation laws. Numer. Math. 79 (1998) 397–425. Zbl0906.65093
  15. [15] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–436. Zbl0697.65068
  16. [16] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comp. 68 (1984) 1025–1055. Zbl0587.65058
  17. [17] H. Tang and G. Warnecke, A note on ( 2 k + 1 ) -point conservative monotone schemes. ESAIM: M2AN 38 (2004) 345–358. Zbl1075.65113

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.