A Convergence Study of Multisubdomain Schwarz Waveform Relaxation for a Class of Nonlinear Problems. (17th August 2015)

• Record Type: Journal Article
• Authors: Zhang, Liping; Wu, Shu-Lin
• Other Names: Kyamakya Kyandoghere Academic Editor.
• Is Part Of: Mathematical problems in engineering. Volume 2015(2015)
• Publication Date: 2015-08-17
• Subjects: Engineering mathematics -- Periodicals; 510.2462
• DOI: 10.1155/2015/612862
• Languages: English
• ISSNs: 1024-123X

Abstract

Abstract : 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∂ t u = ∂ x x u + 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 ≥ 3 andf ′ ( u ) > 0 . In this paper, we try to finish this gap. Precisely, for a specified subdomain numberN, we find that there exists a quantityd max such that convergence of the algorithm on unbounded time domains is guaranteed iff ′ ( u ) ≤ d max ( ∀ u ∈ R ) . The quantityd max depends onN and 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.