Exploration of a sub-cell stabilization scheme for the high-order Lagrangian discontinuous Galerkin hydrodynamic method

摘要

We explored a variant of the sub-cell mesh stabilization (SMS) scheme introduced by Liu et al. in a context of a third-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method for compressible flows on curvilinear meshes in two-dimensional (2D) Cartesian coordinates. With SMS, a cell is decomposed into four quadrilateral sub-cells, which also move in Lagrangian motion. The Riemann velocity at the vertex and surface vertex of the curvilinear element, and the corresponding surface forces, are calculated by solving the same multidirectional approximate Riemann problem. The novelty here lies in that we use the density at the subcell centroid to define the difference between the cell and the sub-cells. The difference between these two density fields is used to correct the stress (pressure) fed into the Riemann solver. Finally, a consistent Riemann solver having the same number of surface segments can be obtained for both types of vertices (i.e., the vertex and surface vertex). Effective limiting strategies are presented that ensure mono-tonicity of the primitive variables with the high-order DG method. This Lagrangian SMS DG hydrodynamic method conserves mass, momentum, and total energy and satisfies the Geometry Conservation Law (GCL). A suite of test problems are calculated to demonstrate the designed order of accuracy of this method, stable mesh motions, and that this Lagrangian SMS DG method preserves cylindrical symmetry on 1D radial flow problems with equal-angle polar curvilinear meshes.