Finite volume maximum principle for hyperbolic scalar problems

Abstract

We present a new formalism to characterize high-order reconstruction algorithms used in finite volume methods. This formalism provides new tools to examine the properties of these methods. Included in this formalism is the notion of admissible reconstruction methods providing concrete statements regarding the satisfaction of the maximum principle and positivity preservation properties. We demonstrate that the traditional reconstruction limiting algorithms can be recast in our formalism, thus providing new proofs of the maximum principle.