Two-dimensional solutions of MHD equations with an adapted Roe method

In this paper a higher-order Godunov method for two-dimensional solutions of the ideal MHD (magnetohydrodynamic) equations is presented. The method utilizes the finite volume approach with quadrilateral cells. In Section 2 the MHD equations (including flux and source terms) in conservative form are given. The momentum flux is rearranged such that while a source vector is produced, the eigenstructure of the Jacobian matrix does not change. This rearrangement allows a full Roe averaging of the density, velocity and pressure for any value of adiabatic index (contrary to Brio and Wu's conclusion (J. Comput. Phys., 75, 400 (1988)). Full Roe averaging for the magnetic field is possible only when the normal gradient of the magnetic field is negligible; otherwise an arithmetic averaging can be used. This new procedure to get Roe-averaged MHD fields at the interfaces between left and right states has been presented by Asian (Ph.D. Thesis, University of Michigan, 1993; Int. j. numer. methods fluids, 22, 569-580 (1996)). This section also includes the shock structure and an eigensystem for MHD problems. The eigenvalues, right eigenvectors and wave strengths for MHD are given in detail to provide the reader with a full description. The second-order, limited finite volume approach which utilizes quadrilateral cells is given in full detail in Section 3. Section 4 gives one- and two-dimensional numerical results obtained from this method. Finally, conclusions are given in Section 5.