On the TVD property of second order methods for 2D scalar conservation laws
arXiv:2110.00067
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 \cite{Goodman} 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.
The paper is being withdrawn because one of the main statements concerning the TVD property of second-order methods for two-dimensional scalar conservation laws lacks sufficient conditions. A counterexample has been found showing that the claimed TVD property does not hold under the conditions stated in the paper. This affects the validity of the corresponding result and its conclusions