A high-order arbitrary Lagrangian-Eulerian discontinuous Galerkin method for compressible flows in two-dimensional Cartesian and cylindrical coordinates

In this paper, a high-order direct arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme is developed for compressible fluid flows in two-dimensional (2D) Cartesian and cylindrical coordinates. The scheme in 2D cylindrical coordinates is based on the control volume approach and it can preserve the conservation property for all the conserved variables including mass, momentum and total energy. In this hydrodynamic scheme, a kind of high-order Taylor expansion basis function on the general element is used to construct the interpolation polynomials of the physical variables for the DG discretization. The terms including the material derivatives of the test functions are omitted, which simplifies the scheme significantly. Furthermore, the mesh velocity in the direct ALE framework is obtained by implementing an adaptive mesh movement method with a kind of dimensional-splitting type monitor function. This type of mesh movement method can automatically concentrate the mesh nodes near the regions with large gradients of the variables, which can greatly improve the resolutions of numerical solutions near the specified regions. For removing the numerical oscillations in the simulations, a Hermite Weighted Essential Non-oscillatory (HWENO) reconstruction is employed as a slope limiter. Finally, some test cases are displayed to verify the accuracy and the good performance of our scheme.