General element shapes within a tensor-product higher-order space-time discontinuous-Galerkin formulation

A tensor-product higher-order space-time discontinuous-Galerkin method is extended to unstructured element shapes. The use of a tensor-product formulation is key to maintaining efficiency at high polynomial orders. The discrete system of equations arising at each space-time slab is solved using a Jacobian-free Newton-Krylov scheme. An alternatingdirection-implicit (ADI) preconditioner for hexahedra is extended to prisms, pyramids and tetrahedra by solving on the tensor-product space corresponding to the quadrature points on the reference cube. Numerical results demonstrate the ADI preconditioner is able to reduce the stiffness associated with high polynomial orders for a scalar advection problem. A diagonalized variant of the ADI preconditioner for the compressible Navier-Stokes equations is used to perform simulations of the Taylor-Green vortex problem. Numerical results demonstrate the efficiency of higher-order methods for the simulation of compressible turbulent flows.