Premium
Relation between PVM schemes and simple Riemann solvers
Author(s) -
Morales de Luna Tomás,
Castro Díaz Manuel J.,
Parés Carlos
Publication year - 2014
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21871
Subject(s) - mathematics , partial derivative , riemann problem , riemann solver , partial differential equation , polynomial , riemann hypothesis , converse , simple (philosophy) , solver , relation (database) , function (biology) , numerical analysis , property (philosophy) , extension (predicate logic) , mathematical analysis , computer science , mathematical optimization , finite volume method , geometry , physics , database , programming language , philosophy , epistemology , evolutionary biology , biology , mechanics
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 30: 1315–1341, 2014