Relation between PVM schemes and simple Riemann solvers (Q2875718)

Approximate Riemann solvers (ARS) and polynomial viscosity matrix (PVM) methods constitute two general frameworks to derive numerical schemes for hyperbolic systems of partial differential equations. In this work, the relation between these two frameworks is analyzed: The authors show that every PVM method can be interpreted in terms of an approximate Riemann solver provided that it is based on a polynomial that interpolates the absolute value function at some points. Furthermore, the converse is true provided that the ARS satisfies a technical property to be specified. Besides its theoretical interest, this relation provides a useful tool to investigate the properties of some well-known numerical methods that are particular cases of PVM methods, as the analysis of some properties is easier for ARS methods. The usefulness by analyzing the positivity-preservation property is illustrated on numerical examples for the shallow water system.