Multidimensional upwind residual distribution schemes are extended to the context of continuous linear space-time ÿnite elements for the time accurate solution of scalar and hyperbolic systems of conservation laws. The formulation leads to a consistent discretization of the space-time domain, thus retaining the properties of the underlying basic schemes both in space and time. We propose a particular space -time mesh conÿguration containing two layers of elements and three levels of nodes in time. This construction leads to unconditionally stable implicit time stepping while retaining second-order spatial and temporal accuracy in smooth ows and monotone solution across steep gradients. The presented schemes have a strong potential in the ÿeld of moving grids, since they allow a dynamic change of the space-time mesh geometry. Numerical results demonstrate the robustness, accuracy and monotonicity of the method.
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.
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