Ninth MSU-UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 20 (2013), pp. 53-63.

Total variation stability and second-order accuracy at extrema

Ritesh Kumar Dubey

It is well known that high order total variation diminishing (TVD) schemes for hyperbolic conservation laws degenerate to first-order accuracy, even at smooth extrema; hence they suffer from clipping error. In this work, TVD bounds on representative three-point second-order accurate schemes are given for the scalar case, which show that it is possible to obtain second order TVD approximation at points of extrema as well as in steep gradient regions. These bounds can be used to improve existing high order TVD schemes and to reduce clipping error. In a 1D scalar test cases, an existing limiters based high order TVD scheme is applied, along with these second-order schemes using their TVD bounds to show improvement in the numerical results at extrema and steep gradient regions.

Published October 31, 2013.
Math Subject Classifications: 35L65, 65M06, 65M12.
Key Words: Second order accurate schemes; total variationdiminishing schemes; smoothness parameter; hyperbolic equations.

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Ritesh Kumar Dubey
Research Institute, SRM University
Tamilnadu, India

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