Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws (Communications in Computational Physics, 2018)
An entropy stable fully discrete shock capturing space-time DiscontinuousGalerkin (DG) method was proposed in a recent paper [20] to approximate hyperbolicsystems of conservation laws. This numerical scheme involves the solution of avery large nonlinear system of algebraic equations, by a Newton-Krylov method, atevery time step. In this paper, we design efficient preconditioners for the large, nonsymmetriclinear system, that needs to be solved at every Newton step. Two sets ofpreconditioners, one of the block Jacobi and another of the block Gauss-Seidel type aredesigned. Fourier analysis of the preconditioners reveals their robustness and a largenumber of numerical experiments are presented to illustrate the gain in efficiency thatresults from preconditioning. The resulting method is employed to compute approximatesolutions of the compressible Euler equations, even for very high CFL numbers.