Lagrange–Schwarz Waveform Relaxation domain decomposition methods for linear and nonlinear quantum wave problems

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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An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations

Numerische Mathematik ◽

10.1007/s00211-017-0897-3 ◽

2017 ◽

Vol 137 (4) ◽

pp. 923-958 ◽

Cited By ~ 5

Author(s):

X. Antoine ◽

E. Lorin

Keyword(s):

Domain Decomposition ◽

Decomposition Methods ◽

Waveform Relaxation ◽

Domain Decomposition Methods ◽

Imaginary Time ◽

Schwarz Waveform Relaxation ◽

Time Linear

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Schwarz Waveform Relaxation for Heat Equations with Nonlinear Dynamical Boundary Conditions

Abstract and Applied Analysis ◽

10.1155/2013/474608 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7

Author(s):

Shu-Lin Wu

Keyword(s):

Boundary Conditions ◽

Domain Decomposition ◽

Domain Decomposition Method ◽

Linear Convergence ◽

Decomposition Methods ◽

Waveform Relaxation ◽

Heat Equations ◽

Nonlinear Dynamical ◽

Schwarz Waveform Relaxation ◽

Dynamical Boundary Conditions

We are interested in solving heat equations with nonlinear dynamical boundary conditions by using domain decomposition methods. In the classical framework, one first discretizes the time direction and then solves a sequence of state steady problems by the domain decomposition method. In this paper, we consider the heat equations at spacetime continuous level and study a Schwarz waveform relaxation algorithm for parallel computation purpose. We prove the linear convergence of the algorithm on long time intervals and show how the convergence rate depends on the size of overlap and the nonlinearity of the nonlinear boundary functions. Numerical experiments are presented to verify our theoretical conclusions.

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A Convergence Study of Multisubdomain Schwarz Waveform Relaxation for a Class of Nonlinear Problems

Mathematical Problems in Engineering ◽

10.1155/2015/612862 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11

Author(s):

Liping Zhang ◽

Shu-Lin Wu

Keyword(s):

Nonlinear Problems ◽

Decomposition Methods ◽

Time Dependent ◽

Waveform Relaxation ◽

Domain Decomposition Methods ◽

Schwarz Waveform Relaxation ◽

Time Domains ◽

Theoretical Predictions ◽

New Type ◽

Convergence Study

Schwarz waveform relaxation (SWR) is a new type of domain decomposition methods, which is suited for solving time-dependent PDEs in parallel manner. The number of subdomains, namely,N, has a significant influence on the convergence rate. For the representative nonlinear problem∂tu=∂xxu+f(u), convergence behavior of the algorithm in the two-subdomain case is well-understood. However, for the multisubdomain case (i.e.,N≥3), the existing results can only predict convergence whenf′(u)≤0 (∀u∈R). Therefore, there is a gap betweenN≥3andf′(u)>0. In this paper, we try to finish this gap. Precisely, for a specified subdomain numberN, we find that there exists a quantitydmaxsuch that convergence of the algorithm on unbounded time domains is guaranteed iff′(u)≤dmax (∀u∈R). The quantitydmaxdepends onNand we present concise formula to calculate it. We show that the analysis is useful to study more complicated PDEs. Numerical results are provided to support the theoretical predictions.

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