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Preprint 2013-24

Aníbal Coronel, Patricio Cumsille, Mauricio Sepúlveda:

Convergence of a level-set algorithm in scalar conservation laws


This paper is concerned with the convergence of the level-set algorithm introduced by Aslam (2001, J. Comput. Phys. 167, 413-438) for tracking the discontinuities in scalar conservation laws in the case of linear or strictly convex flux function. The numerical method is deduced by an appropriate level-set representation of the entropy solution: the zero of a level-set function is used as an indicator of the discontinuity curves and two auxiliary states, which are assumed continuous trough the discontinuities, are introduced. We rewrite the numerical level-set algorithm as a procedure consisting of three big steps: initialization, evolution and reconstruction. In the initialization step we choose an entropy admissible level-set representation of the initial condition. In the evolution step we solve at each iteration step an uncoupled system of three equations (two conservation laws for the auxiliary states and the level-set equation for the approximation of the level set function) and select the entropy admissible level-set representation of the solution profile at the end of the level time iteration, which is used as the initial condition by the next iteration. The reconstruction is given naturally by the recuperation of the entropy solution by using the level-set representation with auxiliary states and the level-set function determined at the evolution step. We prove the convergence of the numerical solution to the entropy solution in Lp-loc for every p greater than or equal to 1, using L-infinity weak BV estimates and a cell entropy inequality. In addition, some numerical examples focused on the elementary wave interaction are presented.

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This preprint gave rise to the following definitive publication(s):

Aníbal CORONEL, Patricio CUMSILLE, Mauricio SEPúLVEDA: Convergence of a level-set algorithm in scalar conservation laws. Numerical Methods for Partial Differential Equations, vol. 31, 4, pp. 1310-1343, (2015).