An Efficient Implicit Discontinuous Spectral Galerkin Method

The present paper discusses an implicit discontinuous spectral Galerkin method for the solution of the compressible Euler equations. A matrix-free Newton–Krylov–Schwarz algorithm with one-level and two-level nonoverlapping Schwarz preconditioners is used to solve the implicit systems. The study shows that this method is a factor of 50 faster than an explicit method that employs local time-stepping to accelerate convergence to steady-state solution. Procedures using LU-SGS preconditioner appear to provide the best performance. The two-level procedure is found necessary for relatively fast convergence in the case of large numbers of mesh elements.