Adaptive mesh reconstruction: Total Variation Bound

TL;DR

This paper develops a model to analyze how adaptive mesh reconstruction can control oscillations in numerical schemes for scalar conservation laws, providing bounds on total variation and demonstrating TV decrease over time.

Contribution

It introduces a model for the evolution of solution extrema on adaptive meshes and proves that proper mesh reconstruction can bound and reduce total variation in numerical schemes.

Findings

Proper mesh reconstruction controls oscillations.

Total Variation bounds are established.

TV decreases over time under certain conditions.

Abstract

We consider 3-point numerical schemes for scalar Conservation Laws, that are oscillatory either to their dispersive or anti-diffusive nature. Oscillations are responsible for the increase of the Total Variation (TV); a bound on which is crucial for the stability of the numerical scheme. It has been noticed (\cite{Arvanitis.2001}, \cite{Arvanitis.2004}, \cite{Sfakianakis.2008}) that the use of non-uniform adaptively redefined meshes, that take into account the geometry of the numerical solution itself, is capable of taming oscillations; hence improving the stability properties of the numerical schemes. In this work we provide a model for studying the evolution of the extremes over non-uniform adaptively redefined meshes. Based on this model 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 behaviour of the numerical schemes- decreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI-D).