An Efficient Low Memory Implicit DG Algorithm for Time Dependent Problems

We present an efficient implicit time stepping method for Discontinuous Galerkin discretizations of the compressible Navier-Stokes equations on unstructured meshes. The Local Discontinuous Galerkin method is used for the discretization of the viscous terms. For unstructured meshes, the Local Discontinuous Galerkin method is known to produce non-compact discretizations. In order to circumvent the difficulties accociated with this non-compactness, we represent the irregular matrices arising from the discretization algorithm as a product of matrices with a more structured pattern. Time integration is carried out using backward difference formulas. This leads to a non-linear system of equations to be solved at each timestep. In this paper, we study various iterative solvers for the linear systems of equations that arise in the Newton algorithm. We show that a two-level preconditioner with incomplete LU as a pre-smoother is highly efficient yet inexpensive to compute and to store. It performs particularly well for low Mach number flows, where it is more than a magnitude more efficient than pure two-level or ILU preconditioning. Our methods are demonstrated using three typical test problems with various parameters and timesteps.