Tensor-product preconditioners for higher-order space–time discontinuous Galerkin methods

A space–time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier–Stokes equations. An efficient solution technique based on a matrix-free Newton–Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high-order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space–time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier–Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows.