Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Mehmet Ersoy; Frédéric Golay; Lyudmyla Yushchenko

Open Mathematics (2013)

  • Volume: 11, Issue: 8, page 1392-1415
  • ISSN: 2391-5455

Abstract

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We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod’s shock tube problem, Lax’s shock tube problem and the Shu-Osher test problem.

How to cite

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Mehmet Ersoy, Frédéric Golay, and Lyudmyla Yushchenko. "Adaptive multiscale scheme based on numerical density of entropy production for conservation laws." Open Mathematics 11.8 (2013): 1392-1415. <http://eudml.org/doc/269362>.

@article{MehmetErsoy2013,
abstract = {We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod’s shock tube problem, Lax’s shock tube problem and the Shu-Osher test problem.},
author = {Mehmet Ersoy, Frédéric Golay, Lyudmyla Yushchenko},
journal = {Open Mathematics},
keywords = {Hyperbolic system; Finite volume scheme; Local mesh refinement; Numerical density of entropy production; Local time stepping; hyperbolic system; finite volume scheme; local mesh refinement; numerical density of entropy production; local time stepping},
language = {eng},
number = {8},
pages = {1392-1415},
title = {Adaptive multiscale scheme based on numerical density of entropy production for conservation laws},
url = {http://eudml.org/doc/269362},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Mehmet Ersoy
AU - Frédéric Golay
AU - Lyudmyla Yushchenko
TI - Adaptive multiscale scheme based on numerical density of entropy production for conservation laws
JO - Open Mathematics
PY - 2013
VL - 11
IS - 8
SP - 1392
EP - 1415
AB - We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod’s shock tube problem, Lax’s shock tube problem and the Shu-Osher test problem.
LA - eng
KW - Hyperbolic system; Finite volume scheme; Local mesh refinement; Numerical density of entropy production; Local time stepping; hyperbolic system; finite volume scheme; local mesh refinement; numerical density of entropy production; local time stepping
UR - http://eudml.org/doc/269362
ER -

References

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