An efficient preconditioner for evolutionary partial differential equations with $\theta$-method in time discretization

TL;DR

This paper introduces a novel parallel-in-time preconditioner for evolutionary PDEs discretized with the $ heta$-method, ensuring efficient GMRES convergence regardless of problem size.

Contribution

The study proposes a new one-sided preconditioner for all-at-once systems from PDE discretization, with theoretical guarantees and parallel implementation advantages.

Findings

Preconditioner ensures GMRES convergence independent of matrix size.

The condition number of the preconditioned matrix is uniformly bounded.

Numerical experiments demonstrate the method's efficiency and robustness.

Abstract

In this study, the $\theta$-method is used for discretizing a class of evolutionary partial differential equations. Then, we transform the resultant all-at-once linear system and introduce a novel one-sided preconditioner, which can be fast implemented in a parallel-in-time way. By introducing an auxiliary two-sided preconditioned system, we provide theoretical insights into the relationship between the residuals of the generalized minimal residual (GMRES) method when applied to both one-sided and two-sided preconditioned systems. Moreover, we show that the condition number of the two-sided preconditioned matrix is uniformly bounded by a constant that is independent of the matrix size, which in turn implies that the convergence behavior of the GMRES method for the one-sided preconditioned system is guaranteed. Numerical experiments confirm the efficiency and robustness of the proposed preconditioning approach.