Solving wave equations in a time-parallel manner is challenging, and the algorithm based on the block \alpha-circulant preconditioning technique has shown its advantage in many existing studies (where \alpha\in(0, 1) is a free parameter). Considerable efforts have been devoted to exploring the  spectral radius of the preconditioned matrix and this leads to many case-by-case studies depending on the used time-integrator. In this paper, we propose a unified way to analyze the convergence via directly studying the error of the algorithm and using the stability of the time integrator. Our analysis works for all^+ with \sigma(A) being the spectrum of A\inC^{m\times m}. For any } one-step time-integrators and two exemplary classes of two-step time-integrators: the parameterized Numerov methodsand the parameterized two-stage hybrid methods. The main conclusion is that the global error satisfies \|{\textbf{err}^{k+1}\|_{{\mathbf W}, 2}\leq \alpha{1-\alpha}\|{err}^{k}\|_{{\mathbf W}, 2} provided that the time-integrator } (z), we prove that the decay rate of the global error is bounded by \alpha/(1-\alpha), provided the method } is stable, where k is the iteration index and for any vector v the norm \|v\|_{{\mathbf W, 2} is defined by \|v\|_{{\mathbf W}, 2}=\|{\mathbf W}v\|_2 with }%DIFDELCMD < {%%% \mathbf W being a matrix depending on the space discretization matrix only. Even though we focus on wave equations in this paper, the proposed argument is directly applicable to other evolution problems|\mathcal{R}(\Delta t\lambda)|}\leq1\geq1  .
Solving evolutionary equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block \alpha-circulant preconditioning technique has shown promising advantages, especially for wave propagation problems. By fast Fourier transform for factorizing the involved circulant matrices, the preconditioned iteration can be computed efficiently via the so-called diagonalization technique, which yields a direct parallel implementation across all time levels. In recent years, considerable efforts have been devoted to exploring the convergence of the preconditioned iteration by studying the spectral radius of the iteration matrix, and this leads to many case-by-case studies depending on the used time-integrator. In this paper, we propose a unified convergence analysis for the algorithm applied to u'+Au=f, where \sigma(A)\subset\mathbb{C^+ with \sigma(A) being the spectrum of A\inC^{m\times m}. For any } one-step }^{k+1}\|_{{\mathbf W}, 2}\leq \alpha{1-\alpha}\|{err}^{k}\|_{{\mathbf W}, 2} provided that the time-integrator } method (such as the Runge-Kutta methods) with stability function \mathcal{R(z), we prove that the decay rate of the global error is bounded by \alpha/(1-\alpha), provided the method } is stable, , 2} is defined by \|v\|_{{\mathbf W}, 2}=\|{\mathbf W}v\|_2 with }%DIFDELCMD < {%%% i.e., \max_{\lambda\in\sigma(A)|\mathcal{R}(\Delta t\lambda)|}\leq1. For any linear multistep method, such a bound becomes c\alpha/(1-c\alpha), where c\geq1 is a constant specified by the multistep method itself. Our proof only relies on the stability of the time-integrator and the estimate is independent of the step size \Delta t and the spectrum \sigma(A) .
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wave
evolutionary
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time-parallel manner is challenging, and the
parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The
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its advantage in many existing studies (where \alpha\in(0, 1) is a free parameter). Considerable
promising advantages, especially for wave propagation problems. By fast Fourier transform for factorizing the involved circulant matrices, the preconditioned iteration can be computed efficiently via the so-called diagonalization technique, which yields a direct parallel implementation across all time levels. In recent years, considerable
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convergence of the preconditioned iteration by studying the
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preconditioned matrix
iteration matrix,
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way to analyze the convergence via directly studying the error of the algorithm and using the stability of the time integrator. Our analysis works for all
convergence analysis for the algorithm applied to u'+Au=f, where \sigma(A)\subset\mathbb{C
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time-integrators and two exemplary classes of two-step time-integrators: the parameterized Numerov methodsand the parameterized two-stage hybrid methods. The main conclusion is that the global error satisfies \|{\textbf{err
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method (such as the Runge-Kutta methods) with stability function \mathcal{R
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where k is the iteration index and for any vector v the norm \|v\|_{{\mathbf W
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\mathbf W
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being a matrix depending on the space discretization matrix only. Even though we focus on wave equations in this paper, the proposed argument is directly applicable to other evolution problems
i.e., \max_{\lambda\in\sigma(A)
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. For any linear multistep method, such a bound becomes c\alpha/(1-c\alpha), where c
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is a constant specified by the multistep method itself. Our proof only relies on the stability of the time-integrator and the estimate is independent of the step size \Delta t and the spectrum \sigma(A)