A characteristic-based nonconvex entropy-fix upwind scheme for the ideal magnetohydrodynamic equations

In this paper we perform an analysis of the wave structure of the ideal magnetohydrodynamic (MHD) equations. We present an analytical expression of the nonlinearity term associated to each characteristic field derived from a scaled version of the complete system of eigenvectors proposed by Brio and Wu [M. Brio, C.C. Wu, An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. Comput. Phys. 75 (2) (1988) 400–422] and adopting the eight wave approach by Powell et al. [K.G. Powell, P.L. Roe, R.S. Myong, T. Gombosi, D. deZeeuw, An upwind scheme for magnetohydrodynamics, AIAA 12th Computational Fluid Dynamics Conference, San Diego, CA, 1995, pp. 661–674]. A criterion for the detection of local regions containing points for which a nonlinear characteristic field becomes nonconvex is formulated for the two-dimensional case. We then design a characteristic-based upwind scheme for the ideal MHD equations that resolves the wave dynamics by local characteristic wavefields. The new scheme is able to detect local regions containing nonconvex singularities and to handle an entropy correction through a prescription of a local viscosity ensuring convergence to the entropy solution. A third order accurate version of the scheme performs satisfactorily in resolving one and two-dimensional MHD problems. Numerical results indicate that the proposed scheme behaves low dissipative, stable and accurate under high CFL numbers.