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

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.

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

2018 ◽  
Vol 52 (4) ◽  
pp. 1569-1596 ◽  
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.

scholarly journals Schwarz Waveform Relaxation for Heat Equations with Nonlinear Dynamical Boundary Conditions

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.

Domain Decomposition Methods With Overlapping Subdomains For The Time-Dependent Problems Of Mathematical Physics

2008 ◽  
Vol 8 (4) ◽  
pp. 393-405 ◽  
Author(s):  
P.N. VABISHCHEVICH
Keyword(s):  
Domain Decomposition ◽  
Data Exchange ◽  
Mathematical Physics ◽  
Decomposition Methods ◽  
Time Dependent ◽  
Parallel Computer ◽  
Approximation Schemes ◽  
Domain Decomposition Methods ◽  
Additive Schemes ◽  
Time Dependent Problems

AbstractAt the present time, the domain decomposition methods are considered as the most promising ones for parallel computer systems. Nowadays success is attained mainly in solving approximately the classical boundary value problems for second-order elliptic equations. As for the time-dependent problems of mathematical physics, there are, in common use, approaches based on ordinary implicit schemes and implemented via iterative methods of the domain decomposition. An alternative technique is based on the non-iterative schemes (region-additive schemes). On the basis of the general theory of additive schemes a wide class of difference schemes (alternative directions, locally one-dimensional, factorized schemes, summarized approximation schemes, vec-tor additive schemes, etc.) as applied to the domain decomposition technique for time-dependent problems with synchronous and asynchronous implementations has been investigated. For nonstationary problems with self-adjoint operators, we have considered three dif-ferent types of decomposition operators corresponding to the Dirichlet and Neumann conditions on the subdomain boundaries. General stability conditions have been obtained for the region-additive schemes. We focused on the accuracy of domain decom-position schemes. In particular, the dependence of the convergence rate on the width of subdomain overlapping has been investigated as the primary property. In the present paper, new classes of domain decomposition schemes for nonstationary problems, based on the subdomain overlaping and minimal data exchange in solving problems in subdomains, have been constructed.

Sign in / Sign up

Export Citation Format

Share Document