A Fast Block $\alpha$-Circulant Preconditoner for All-at-Once Systems From Wave Equations

Abstract

In this paper, we propose a fast block $\alpha$-circulant preconditioner for solving the nonsymmetric linear system arising from an all-at-once implicit discretization scheme in time for the wave equation. As a generalization of the well-known block circulant preconditioning technique, the proposed block $\alpha$-circulant preconditioner can also be efficiently inverted in a parallel-in-time manner. The complex eigenvalues of the preconditioned matrix are fully derived in explicit expression and its diagonalizability is also shown. Furthermore, a mesh-independent convergence rate of the preconditioned GMRES method is derived under certain conditions. Building upon the proposed preconditioner, two stationary iterative methods with uniform asymptotic convergence rates were also presented. The extension of our preconditioner within a simplified Newton iteration to the nonlinear wave equation is also discussed. Both linear and nonlinear numerical examples are given to illustrate the promising performance of our proposed block $\alpha$-circulant preconditioner and also validate our theoretical results on convergence analysis.