Asymptotic Results of Schwarz Waveform Relaxation Algorithm for Time Fractional Cable Equations

Heat equations with distributed delay are a class of mathematic models that has wide applications in many fields. Numerical computation plays an important role in the investigation of these equations, because the analytic solutions of partial differential equations with time delay are usually unavailable. On the other hand, duo to the delay property, numerical computation of these equations is time-consuming. To reduce the computation time, we analyze in this paper the Schwarz waveform relaxation algorithm with Robin transmission conditions. The Robin transmission conditions contain a free parameter, which has a significant effect on the convergence rate of the Schwarz waveform relaxation algorithm. Determining the Robin parameter is therefore one of the top-priority matters for the study of the Schwarz waveform relaxation algorithm. We provide new formula to fix the Robin parameter and we show numerically that the new Robin parameter is more efficient than the one proposed previously in the literature.

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A parallel Aitken-additive Schwarz waveform relaxation suitable for the grid

Parallel Computing ◽

10.1016/j.parco.2009.05.001 ◽

2009 ◽

Vol 35 (7) ◽

pp. 416-428 ◽

Cited By ~ 3

Author(s):

Hatem Ltaief ◽

Marc Garbey

Keyword(s):

Waveform Relaxation ◽

Additive Schwarz ◽

Schwarz Waveform Relaxation

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Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2017048 ◽

2018 ◽

Vol 52 (4) ◽

pp. 1569-1596 ◽

Cited By ~ 4

Author(s):

Xavier Antoine ◽

Fengji Hou ◽

Emmanuel Lorin

Keyword(s):

Domain Decomposition ◽

Decomposition Methods ◽

Nonlinear Schrödinger Equations ◽

Waveform Relaxation ◽

Schrödinger Equations ◽

Two Dimensional ◽

Domain Decomposition Methods ◽

Schwarz Waveform Relaxation ◽

Nonlinear Schrodinger Equations ◽

Linear And Nonlinear

This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [X. Antoine and E. Lorin, Numer. Math. 137 (2017) 923–958] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

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