A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

In this work, we present a class of fast first order finite volume solvers, called PVM (polynomial viscosity matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of a Roe matrix is needed. As a consequence, they are faster than the Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [C. R. Acad. Sci. Paris Ser . l Math ., 328 (1999), pp. 479--483] for balance laws and nonconservative systems. The first order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multidimensional problems. In this work, some well known solvers, such as Rusanov, Lax--Friedrichs, FORCE (see [E. F. Toro and S. J. Billett, IMA J. Numer. Anal ., 20 (2000), pp. 47--79; M. J. Castro, A. Pardo, C. Pares, and E. F. Toro, Math. Comp ., 79 (2010), pp. 1427--1472]), GFORCE (see [E. F. Toro and V. A. Titarev, J. Comput. Phys ., 216 (2006), pp. 403--429]), and HLL (see [A. Harten, P. D. Lax, and B. van Leer,SIAM Rev ., 25 (1983), pp. 35--61]), are redefined under this form, and then some new solvers are proposed. Finally, some numerical tests are presented, and the performances of the numerical schemes are compared with each others and with Roe scheme for the 1D and 2D two-fluid flow model of Pitman and Le.