Theoretical and computational investigations of nonlinear waves in magnetohydrodynamics.

For the purpose of developing a basic theory of the nonlinear evolution and stability of MHD waves and an approximate Riemann solver for MHD, the Riemann problem is investigated theoretically and computationally. The investigation starts with an analysis of a model system that preserves exactly the MHD singularity. Shock admissibility conditions are extensively examined and it is shown that the simple geometric conditions are inappropriate for determining physically relevant shocks of non-strictly hyperbolic conservation laws. A viscosity admissibility condition is proposed that ensures the uniqueness of the Riemann problem, and a global analysis of the dynamical system of the model is presented. By observing that the MHD system is symmetrically hyperbolic, a new MHD Rankine-Hugoniot relation is derived and then connected with recent mathematical theories of the Riemann problem of a quadratic model system near a hyperbolic singularity. As a result, a proof of the well-posedness of the planar MHD Riemann problem is obtained. By studying the nonlinear evolution of the finite-amplitude waves, it is shown that the solution to the non-planar Riemann problem is not necessarily unique, in particular, that the solutions depend not only on left and right states, but also on the associated magnetic field moments. Finally, some numerical issues in developing Godunov-type numerical methods for MHD are discussed and a new approximate MHD Riemann solver is proposed, which is remarkably simple and follows exactly the way to generate the large-time solution to the non-planar Riemann problem from the well-posed planar problem.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105128/1/9635579.pdfDescription of 9635579.pdf : Restricted to UM users only