Stationary Two-Dimensional Magnetohydrodynamic Flows with Shocks: Characteristic Analysis and Grid Convergence Study

Five model flows of increasing complexity belonging to the class of stationary two-dimensional planar field-aligned magnetohydrodynamic (MHD) flows are presented which are well suited to the quantitative evaluation of MHD codes. The physical properties of these five flows are investigated using characteristic theory. Grid convergence criteria for flows belonging to this class are derived from characteristic theory, and grid convergence is demonstrated for the numerical simulation of the five model flows with a standard high-resolution finite volume numerical MHD code on structured body-fitted grids. In addition, one model flow is presented which is not field-aligned, and it is discussed how grid convergence can be studied for this flow. By formal grid convergence studies of magnetic flux conservation and other flow quantities, it is investigated whether the Powell source term approach to controlling the ∇·B constraint leads to correct results for the class of flows under consideration.