A TVD Active Flux method for scalar conservation laws

Abstract

Focusing on scalar conservation laws, a TVD theory is developed for the fully discrete, single-stage Active Flux method. The classical choice of degrees of freedom is considered, yielding a third-order method. Starting from an adaptation of the concept of (discrete) total variation to Active Flux, a corresponding adaptation of Harten's lemma is achieved. Finally, the high-order method is endowed with parameters chosen such that they can be efficiently computed to fulfill the conditions of the lemma.