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An Efficient Preconditioner for Evolutionary Partial Differential Equations with θ-Method in Time Discretization
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DOI：10.1007/s10915-025-02862-9.png)
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Abstract
En 中文
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.
Keywords:
All-at-once systems
theta-Method
Preconditioning
Multilevel Toeplitz matrices
Crank-Nicolson
Parallel-in-time
Journal
IF:
3.3
Papers:
680
Citations:
9.6K