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Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound


Author: Nikolaos Sfakianakis
Journal: Math. Comp. 82 (2013), 129-151
DOI: https://doi.org/10.1090/S0025-5718-2012-02615-9
Published electronically: August 16, 2012
MathSciNet review: 2983018
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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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Additional Information

Nikolaos Sfakianakis
Affiliation: University of Vienna, Austria
Address at time of publication: Johannes Gutenberg University, Mainz, Germany
Email: sfakiana@uni-mainz.de

DOI: https://doi.org/10.1090/S0025-5718-2012-02615-9
Received by editor(s): September 19, 2009
Received by editor(s) in revised form: September 2, 2011
Published electronically: August 16, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.