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

•An efficient implementation of PVM methods is proposed.•The relation between the PVM methods and simple Riemann solvers is revisited.•A new implementation of the Roe method is proposed, named as Newton Roe method.•Newton Roe method is more efficient than the standard one for large hyperbolic systems.•Numerical tests with the two-layer shallow-water system and the QBME model are shown. 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.