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Convergence analysis of a periodic-like waveform relaxation method for initial-value problems via the diagonalization technique

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Abstract

We present in this paper a time parallel algorithm for \({\dot{u}}=f(t,u)\) with initial-value \(u(0)=u_0\), by using the waveform relaxation (WR) technique, and the diagonalization technique. With a suitable parameter \(\alpha \), the WR technique generates a functional sequence \(\{u^k(t)\}\) via the dynamic iterations \({\dot{u}}^k=f(t,u^k)\), \(u^k(0)=\alpha u^k(T)-\alpha u^{k-1}(T)+u_0\), and at convergence we get \(u^{\infty }(t)=u(t)\). Each WR iterate represents a periodic-like differential equation, which is very suitable for applying the diagonalization technique yielding direct parallel-in-time computation. The parameter \(\alpha \) controls both the roundoff error arising from the diagonalization procedure and the convergence factor of the WR iterations, and we perform a detailed analysis for the influence of the parameter \(\alpha \) on the method. We show that the roundoff error is proportional to \(\epsilon (2N+1)\max \{|\alpha |^2, |\alpha |^{-2}\}\) (\(N=T/\varDelta t\) and \(\epsilon \) is the machine precision), and the convergence factor can be bounded by \(|\alpha |e^{-TL}/(1-|\alpha |e^{-TL})\), where \(L\ge 0\) is the one-sided Lipschitz constant of f. We also perform a convergence analysis at the discrete level and the effect of temporal discretizations is explored. Our analysis includes the heat and wave equations as special cases. Numerical results are given to support our findings.

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Notes

  1. For the Backward-Euler method \({\tilde{\rho }}=\tilde{\rho _1}\) and for the Trapezoidal rule \({\tilde{\rho }}={\tilde{\rho }}_{\frac{1}{2}}\), where \({\tilde{\rho }}_1\) and \({\tilde{\rho }}_{\frac{1}{2}}\) are given by Theorems 3.2 and 3.3.

  2. This choice may be not optimal in practice.

  3. According to [18, Section 5], the communication cost for the parallel implementation of FFT is negligible.

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Acknowledgements

The authors are very grateful to the anonymous referees for their careful reading of a preliminary version of the manuscript and their valuable suggestions, which greatly improved the quality of this paper. The second author is supported by the NSF of China (No. 11771313).

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Appendix

Appendix

The main notations and symbols of this paper are listed in the following table.

A

Coefficient matrix

\(\theta \)

Linear \(\theta \)-method (\(\theta =1\) or \(\theta =\frac{1}{2}\))

T

Length of time interval

\(\varDelta t\)

Time step-size

N

Number of time steps

\(\varDelta x\)

Space mesh-size

m

Dimension of IVP

Cond(\(\cdot \))

Condition number of a matrix

\(\omega \)

Opening angle of sector

L

Lipschitz constant

\(\mu \)

Eigenvalue of A

\(\varepsilon \)

Tolerance for WR iterations

\(\eta _0\)

Minimal real part of \(\mu (A)\)

\(\epsilon \)

Machine precision

\(\eta \)

\(\eta =\eta _0T\)

\(\rho \)

Convergence factor (continuous)

\({\tilde{\eta }}\)

\({\tilde{\eta }}=\eta _0\varDelta t\)

\({\tilde{\rho }}\)

Convergence factor (discrete)

\(\alpha \)

Parameter used in WR iteration

\(I_t\)

\(I_t\in {\mathbb {R}}^{N\times N}\) is an identity matrix

k

Index of WR iteration

\(I_x\)

\(I_t\in {\mathbb {R}}^{m\times m}\) is an identity matrix

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Gander, M.J., Wu, SL. Convergence analysis of a periodic-like waveform relaxation method for initial-value problems via the diagonalization technique. Numer. Math. 143, 489–527 (2019). https://doi.org/10.1007/s00211-019-01060-8

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