Roe Matrices for Ideal MHD and Systematic Construction of Roe Matrices for Systems of Conservation Laws

Abstract

In this paper, the construction of a Roe's scheme for the conservative system of ideal magnetohydrodynamics (MHD) is presented. As this method relies on the computation of a Roe matrix, the problem is to find a matrixA(Ul,Ur) which satisfies the following properties. It is required to be consistent with the jacobian of the fluxF, to have real eigenvalues, a complete set of eigenvectors and to satisfy the relation: ΔF=A(Ul,Ur) ΔU, whereUlandUrare two admissible states and ΔUtheir difference. For the ideal MHD system, using eulerian coordinates, a Roe matrix is obtained without any hypothesis on the specific heat ratio. Especially, its construction relies on an original expression of the magnetic pressure jump. Moreover, a Roe matrix is computed for lagrangian ideal MHD, by extending the results of Munz who obtained such a matrix for the system of lagrangian gas dynamics. So this second matrix involves arithmetic averages unlike the eulerian one, which contains classical Roe averages like in eulerian gas dynamics. In this paper, a systematic construction of lagrangian Roe matrices in terms of eulerian Roe matrices for a general system of conservation laws is also presented. This result, applied to the above eulerian and lagrangian matrices for ideal MHD, gives two new matrices for this system. In the same way, by applying this construction to the gas dynamics equations new Roe matrices are also obtained. All these matrices allow the construction of Roe type schemes. Some numerical examples on the shock tube problem show the applicability of this method.