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arXiv · 2408.03535

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

Yuan-Yuan Huang·Po Yin Fung·Sean Y. Hon·Xue-Lei Lin
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Abstract

In this study, the $θ$-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.

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Yuan-Yuan Huang, Po Yin Fung, Sean Y. Hon, Xue-Lei Lin. 2024-08-07. An efficient preconditioner for evolutionary partial differential equations with $θ$-method in time discretization. https://arxiv.org/abs/2408.03535

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