Central European Journal of Mathematics

, Volume 11, Issue 8, pp 1392–1415 | Cite as

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

  • Mehmet ErsoyEmail author
  • Frédéric Golay
  • Lyudmyla Yushchenko
Research Article


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.


Hyperbolic system Finite volume scheme Local mesh refinement Numerical density of entropy production Local time stepping 


74S10 35L60 74G15 


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Copyright information

© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  • Mehmet Ersoy
    • 1
    Email author
  • Frédéric Golay
    • 1
  • Lyudmyla Yushchenko
    • 1
  1. 1.IMATH, EA 2134Université de ToulonLa Garde CedexFrance

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