Preprint 2001-053

Discrete Cell Entropy Inequalities for Scalar Conservation Laws

Thorsten Grahs and Thomas Sonar

Abstract: We analyze semi-discrete three-point finite difference schemes for the numerical solution of scalar conservation laws and in particular their behaviour on discrete entropy inequalities. Special emphasize is given to the choice of the discretization of the numerical entropy flux. We show that our simple choice is consistent with a classical approach due to Crandall and Majda. Starting from this discretization of the entropy inequality we derive cell entropy inequalities which enables us to determine the dissipation model of the corresponding numerical method. In the following we show that this is consistent with the dissipation coefficient for the Roe scheme for scalar equations. We show that the necessary entropy fix for the Roe scheme proposed by Harten and Hyman is also necessary by the derivation from cell entropy inequalities.

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Thorsten Grahs, <>
Thomas Sonar, <>
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<> Desember 21 2001.

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