An efficient and robust high-order compact ALE gas-kinetic scheme for unstructured meshes

For the arbitrary-Lagrangian-Eulerian (ALE) calculations, the geometric information needs to be calculated at each time step due to the movement of mesh. To achieve the high-order spatial accuracy, a large number of matrix inversions are needed, which affect the efficiency of computation dramatically. In this paper, an efficient and robust high-order compact ALE gas-kinetic scheme is developed for the compressible moving grids and moving boundary problems. The memory-reduction reconstruction is used to construct a quadratic polynomial on the target cell, where both structured and unstructured meshes can be used. Taking derivatives of the candidate polynomial, the quadratic terms can be obtained by the least square method using the average gradient values of the cell itself and its adjacent cells. Moving the quadratic terms to right-hand side of the constrains for cell averaged value, the linear terms of the polynomial can be determined by the least square method as well. The gradient compression factor is adopted to suppress the spurious oscillations near discontinuities. Combined with the two-stage fourth-order time discretization, a high-order compact gas-kinetic scheme is developed for ALE computation. In the process of mesh movement, the inversions of lower order matrix are needed for the least square method, which makes a 7x speedup and improves the efficiency greatly. In the computation, the grid velocity can be given by the mesh adaptation method and the cell centered Lagrangian nodal solver. Numerical examples are presented to evaluate the accuracy, efficiency, robustness and the preservation of geometric conservation law of the current scheme.