Fast time implicit–explicit discontinuous Galerkin method for the compressible Navier–Stokes equations

An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving the unsteady compressible Navier–Stokes equations. The time discretization is based on an explicit formulation for the convective fluxes and an implicit formulation for the viscous fluxes. The implicit procedure uses a fast iterative algorithm based on a partial uncoupling of variables in neighboring elements introduced in previous works by the authors. In the present work, the method is associated to a Newton–Krylov Jacobian-free method. This association allows to reduce memory requirements and operation counts by avoiding the complete construction of the Jacobian matrix and solving a problem of reduced size. Numerical examples in two space dimensions indicate that the performance of the present method is seen to be significantly improved in terms of CPU time.