ROE MATRICES FOR IDEAL MHD AND SYSTEMATIC CONSTRUCTION OF ROE MATRICES FOR SYSTEMS OF CONSERVATION-LAWS

In this paper, the construction of a Roe's scheme for the conservative system of ideal magnetohydrodynamics (MHD) is presented. As this meth od relies on the computation of a Roe matrix, the problem is to find a matrix A(U-l, U-r) which satisfies the following properties. It is re quired to be consistent with the jacobian of the flux F, to have real eigenvalues, a complete set of eigenvectors and to satisfy the relatio n: Delta F = A(U-l, U-r) Delta U, where U-l and U-r are two admissible states and Delta U their 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 mat rix is computed for lagrangian ideal MHD, by extending the results of Munz who obtained such a matrix for the system of lagrangian gas dynam ics. So this second matrix involves arithmetic averages unlike the eul erian one, which contains classical Roe averages like in eulerian gas dynamics. In this paper, a systematic construction of lagrangian Roe m atrices in terms of eulerian Roe matrices for a general system of cons ervation laws is also presented. This result, applied to the above eul erian and lagrangian matrices for ideal MHD, gives two new matrices fo r this system. In the same way, by applying this construction to the g as dynamics equations new Roe matrices are also obtained. All these ma trices allow the construction of Roe type schemes. Some numerical exam ples on the shock tube problem show the applicability of this method. (C) 1997 Academic Press.