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On the TVD property of second order methods for 2D scalar conservation laws

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On the TVD property of second order methods for 2D scalar conservation laws.pdf (912.01 KB)

Date

2021-10-05

Authors

Krivodonova, Lilia
Smirnov, Alexey

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Publisher

arXiv

Abstract

The total variation diminishing (TVD) property is an important tool for ensuring nonlinear stability and convergence of numerical solutions of one-dimensional scalar conservation laws. However, it proved to be challenging to extend this approach to two-dimensional problems. Using the anisotropic definition for discrete total variation (TV), it was shown in [14] that TVD solutions of two-dimensional hyperbolic equations are at most first order accurate. We propose to use an alternative definition resulting from a full discretization of the semi-discrete Raviart-Thomas TV. We demonstrate numerically using the second order discontinuous Galerkin method that limited solutions of two-dimensional hyperbolic equations are TVD in means when total variation is computed using the new definition

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Keywords

hyperbolic conservation laws, total variation diminishing schemes, discontinuous Galerkin method, high-order methods

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Citation

URI

https://doi.org/10.48550/arXiv.2110.00067
 http://hdl.handle.net/10012/18378

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Waterloo Research
Applied Mathematics