An efficient preconditioner for evolutionary partial differential equations with $\theta$-method in time discretization
<|reference_start|>An efficient preconditioner for evolutionary partial differential equations with $\theta$-method in time discretization: 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.<|reference_end|>
@article{huang2024an,
    title={An efficient preconditioner for evolutionary partial differential
  equations with $\theta$-method in time discretization},
    author={Yuan-Yuan Huang, Po Yin Fung, Sean Y. Hon, Xue-Lei Lin},
    journal={arXiv preprint arXiv:2408.03535},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.03535},
    primaryClass={math.NA cs.NA}
}