Space–Time Discontinuous Galerkin Finite Element Method with Dynamic Grid Motion for Inviscid Compressible Flows: I. General Formulation

A new space–time discontinuous Galerkin finite element method for the solution of the Euler equations of gas dynamics in time-dependent flow domains is presented. The discontinuous Galerkin discretization results in an efficient elementwise conservative upwind finite element method, which is particularly well suited for local mesh refinement. The upwind scheme uses a formulation of the HLLC flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated. The non-linear equations of the space–time discretization are solved using a multigrid accelerated pseudo-time-integration technique with an optimized Runge–Kutta method. The linear stability of the pseudo-time-integration method is investigated for the linear advection equation. The numerical scheme is demonstrated with simulations of the flow field in a shock tube, a channel with a bump, and an oscillating NACA 0012 airfoil. These simulations show that using the data at the superconvergence points, the accuracy of the numerical discretization is O(h5/2) in space for smooth subsonic flows, both on structured and on locally refined meshes, and that the space–time adaptation can significantly improve the accuracy and efficiency of the numerical method.