A Phase Error Analysis of Multigrid Methods for Hyperbolic Equations

In this paper, we study the effects of the coarse grid correction process on multigrid convergence for hyperbolic problems in one and two dimensions. We approach this from the perspective of phase error, which allows us to exploit the hyperbolic nature of the underlying PDE. In particular, we consider three combinations of coarse grid operators and coarse grid solution approaches: (1) inexact coarse grid solve with direct discretization, (2) exact coarse grid solve with direct discretization, and (3) exact coarse grid solve with Galerkin coarse grid operator. For all these approaches, we show that the convergence behavior of multigrid can be precisely described by the phase error analysis of the coarse grid correction matrix, and we verify our results by numerical examples in one and two dimensions.