Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound

Sfakianakis, Nikolaos

doi:10.1090/S0025-5718-2012-02615-9

Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound
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by Nikolaos Sfakianakis;

Math. Comp. 82 (2013), 129-151

DOI: https://doi.org/10.1090/S0025-5718-2012-02615-9

Published electronically: August 16, 2012

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Abstract:

We consider 3-point numerical schemes, that resolve scalar conservation laws, that are oscillatory either to their dispersive or anti-diffusive nature. The spatial discretization is performed over non-uniform adaptively redefined meshes. We provide a model for studying the evolution of the extremes of the oscillations. We prove that proper mesh reconstruction is able to control the oscillations; we provide bounds for the Total Variation (TV) of the numerical solution. We, moreover, prove under more strict assumptions that the increase of the TV, due to the oscillatory behavior of the numerical schemes, decreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI-D). We finally provide numerical evidence supporting the analytical results that exhibit the stabilization properties of the mesh adaptation technique.

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