Analysis of Implicit Time-Advancing p-Multigrid Schemes for Discontinuous Galerkin Discretizations of the Euler Equations

Abstract

We have analyzed various implicit time-advancing (ITA) multigrid schemes for convergence to the steady-state solution of discontinuous Galerkin (DG) discretizations of the Euler equations. This work complements our earlier work 1 where preconditioned approximate inversion (AI) solution methods were analyzed. The discretizations use approximating polynomials of orders p = 0, 2, and 4. For the higher-order discretizations (p = 2, and 4), preconditioned relaxation schemes are coupled with two-level p-multigrid, and for comparison, on the p = 0 discretization the same schemes are coupled with two-level agglomeration multigrid. Relaxation schemes with block -diagonal, -line, and sweeping preconditioners are analyzed and compared with earlier results from the AI methods. In addition, the block alternate direction implicit (ADI) scheme and the incomplete lower-upper factorization (ILU(0)) scheme are analyzed. The method of analysis for the ILU(0) scheme is unique and is described in detail. The results show that in most cases the ITA schemes do not perform as well as AI schemes.