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

TL;DR

This paper investigates the TVD property for second order methods in 2D scalar conservation laws, proposing a new TV definition that enables second order TVD solutions in 2D, supported by numerical evidence.

Contribution

It introduces an alternative total variation definition based on Raviart-Thomas discretization, allowing second order TVD solutions in 2D, overcoming previous limitations.

Findings

Second order solutions can be TVD in mean with the new TV definition.

Numerical results confirm the effectiveness of the proposed approach.

The new TV definition improves stability analysis for 2D hyperbolic equations.

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.