TY - JOUR

T1 - On the efficient implementation of PVM methods and simple Riemann solvers. Application to the Roe method for large hyperbolic systems

AU - Pimentel, Ernesto

AU - Pares, Carlos

AU - Castro, Manuel

AU - Koellermeier, Julian

PY - 2021/1

Y1 - 2021/1

N2 - Polynomial Viscosity Matrix (PVM) methods can be considered as approximations of the Roe method in which the absolute value of the Roe matrix appearing in the numerical viscosity is replaced by the evaluation of the Roe matrix at a chosen polynomial that approximates the absolute value function. They are in principle cheaper than the Roe method since the computation and the inversion of the eigenvector matrix is not necessary. In this article, an efficient implementation of the PVM based on polynomials that interpolate the absolute value function at some points is presented. This implementation is based on the Newton form of the polynomials. Moreover, many numerical methods based on simple Riemann solvers (SRS) may be interpreted as PVM methods and thus this implementation can be also applied to them: the close relation between PVM methods and simple Riemann solvers is revisited here and new shorter proofs based on the classical interpolation theory are given. In particular, Roe method can be interpreted both as a SRS and as a PVM method so that the new implementation can be used. This implementation, that avoids the computation and the inversion of the eigenvector matrix, is called Newton Roe method. Newton Roe method yields the same numerical results of the standard Roe method, with less runtime for large PDE systems. Numerical results for two-layer Shallow Water Equations and Quadrature-Based Moment Equations show a significant speedup if the number of equations is large enough.

AB - Polynomial Viscosity Matrix (PVM) methods can be considered as approximations of the Roe method in which the absolute value of the Roe matrix appearing in the numerical viscosity is replaced by the evaluation of the Roe matrix at a chosen polynomial that approximates the absolute value function. They are in principle cheaper than the Roe method since the computation and the inversion of the eigenvector matrix is not necessary. In this article, an efficient implementation of the PVM based on polynomials that interpolate the absolute value function at some points is presented. This implementation is based on the Newton form of the polynomials. Moreover, many numerical methods based on simple Riemann solvers (SRS) may be interpreted as PVM methods and thus this implementation can be also applied to them: the close relation between PVM methods and simple Riemann solvers is revisited here and new shorter proofs based on the classical interpolation theory are given. In particular, Roe method can be interpreted both as a SRS and as a PVM method so that the new implementation can be used. This implementation, that avoids the computation and the inversion of the eigenvector matrix, is called Newton Roe method. Newton Roe method yields the same numerical results of the standard Roe method, with less runtime for large PDE systems. Numerical results for two-layer Shallow Water Equations and Quadrature-Based Moment Equations show a significant speedup if the number of equations is large enough.

KW - PVM methods

KW - Simple Riemann solvers

KW - Roe method

KW - Finite volume methods

KW - Path-conservative methods

KW - Large hyperbolic systems

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-85088315080&partnerID=MN8TOARS

U2 - 10.1016/j.amc.2020.125544

DO - 10.1016/j.amc.2020.125544

M3 - Article

SN - 0096-3003

VL - 388

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

M1 - 125544

ER -