[en] Recently the shallow water magnetohydrodynamic (SMHD) equations have been proposed for describing the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two-dimensional (2D) with magnetohydrostatic equilibrium in the third direction. In the present paper the properties of the SMHD equations as a nonlinear system of hyperbolic conservation laws are described. Characteristics and Riemann invariants are studied for 1D unsteady and 2D steady flow. Simple wave solutions are derived, and the nonlinear character of the wave modes is investigated. The ∇·(h B)=0 constraint and its role in obtaining a regularized Galilean invariant conservation law form of the SMHD equations is discussed. Solutions of the Rankine-Hugoniot relations are classified and their properties are investigated. The derived properties of the wave modes are illustrated by 1D numerical simulation results of SMHD Riemann problems. A Roe-type linearization of the SMHD equations is given which can serve as a building block for accurate shock-capturing numerical schemes. The SMHD equations are presently being used in the study of the dynamics of layers in the solar interior, but they may also be applicable to problems involving the free surface flow of conducting fluids in laboratory and industrial environments

[en] Recently it has been shown that for strong upstream magnetic field stationary three-dimensional (3D) magnetohydrodynamic (MHD) bow shock flows exhibit a complex double-front shock topology with particular segments of the shock fronts being of the intermediate MHD shock type. The large-scale stability of this new bow shock topology is investigated. It is found that large-amplitude perturbations may cause the disintegration of the intermediate shocks--which are indeed known to be unstable against perturbations with integrated amplitudes above critical values-- but that in the driven bow shock problem there are always shock front segments where intermediate shocks are reformed dynamically, resulting in the reappearance of the new double-front topology. This shows that the new bow shock topology, and shock segments of intermediate type in general, may be found in MHD plasma flows even when there are large-amplitude perturbations

[en] In this paper we study the use of long distance interpolation methods with the low complexity coarsening algorithm PMIS. AMG performance and scalability is compared for classical as well as long distance interpolation methods on parallel computers. It is shown that the increased interpolation accuracy largely restores the scalability of AMG convergence factors for PMIS-coarsened grids, and in combination with complexity reducing methods, such as interpolation truncation, one obtains a class of parallel AMG methods that enjoy excellent scalability properties on large parallel computers