The authors focus on efficiently solving the linear wave equation using a parallel-in-time (PinT) approach. They start by discretizing the wave equation using an implicit leap-frog finite difference scheme, which leads to a large, sparse, and nonsymmetric block Toeplitz linear system.
To accelerate the convergence of the MINRES solver for this system, the authors propose a novel Hermitian positive definite (HPD) preconditioner based on the absolute value of a block α-circulant matrix. The key steps are:
The original block α-circulant preconditioner is not HPD, so it cannot be directly used with the MINRES solver. The authors show how to construct a HPD variant by taking the matrix square root of the block α-circulant matrix.
Theoretical analysis is provided to prove that the MINRES solver with the proposed preconditioner achieves a convergence rate that is independent of the matrix size (i.e., the number of spatial and temporal grid points).
The matrix-vector multiplication with the preconditioner can be efficiently implemented using fast Fourier transforms, enabling fast preconditioning.
Numerical experiments demonstrate the effectiveness of the proposed preconditioner, showing that it outperforms the existing absolute value block circulant preconditioner for wave equations.
The authors emphasize that their work is the first to construct a nontrivial symmetric positive definite version of the block α-circulant preconditioner for the all-at-once system arising from wave equations, filling a gap in the literature.
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