Relation between PVM schemes and simple Riemann solvers. Morales de Luna, T., Castro Díaz, M. J., & Parés, C. 30(4):1315–1341.
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 (PDE's). In this work, the relation between these two frameworks is analyzed: we 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. We illustrate this usefulness by analyzing the positivity-preservation property of some well-known numerical methods for the shallow water system. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2014
@article{morales_de_luna_relation_2014,
title = {Relation between {PVM} schemes and simple Riemann solvers},
volume = {30(4)},
url = {http://onlinelibrary.wiley.com/doi/10.1002/num.21871/abstract},
abstract = {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 ({PDE}'s). In this work, the relation between these two frameworks is analyzed: we 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. We illustrate this usefulness by analyzing the positivity-preservation property of some well-known numerical methods for the shallow water system. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2014},
pages = {1315--1341},
journaltitle = {Numerical Methods for Partial Differential Equations},
author = {Morales de Luna, Tomás and Castro Díaz, Manuel J. and Parés, Carlos},
date = {2014-03},
keywords = {finite volume schemes, {PVM} schemes, Riemann solvers, Shallow water equations},
}