Taiwanese Journal of Mathematics

Embedded Waveform Relaxation Methods for Parabolic Partial Functional Differential Equations

Jun Liu, Yao-Lin Jiang, and Hong-Kun Xu

Full-text: Open access

Abstract

Waveform relaxation methods are decoupling or splitting methods for large scale ordinary differential equations. In this paper, we apply the meth- ods directly to semi-linear parabolic partial functional differential equations. Taking into consideration of the complicated forms of these parabolic equa- tions, we propose a kind of embedded waveform relaxation methods, which are in fact two-level waveform relaxation methods and which can also be ap- plied to some other systems. We provide explicit iterative expressions of the embedded methods and exhibit the superlinear rate of convergence on finite time intervals. We also apply the two-level idea to the functional differential equations derived from semi-discretization of the original system. The win- dowing technique is employed for the situation of long time intervals. Finally, two numerical experiments are performed to confirm our theory.

Article information

Source
Taiwanese J. Math., Volume 15, Number 2 (2011), 829-855.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406237

Digital Object Identifier
doi:10.11650/twjm/1500406237

Mathematical Reviews number (MathSciNet)
MR2810184

Zentralblatt MATH identifier
1234.35293

Subjects
Primary: 35R10: Partial functional-differential equations
Secondary: 35K55: Nonlinear parabolic equations 65M12: Stability and convergence of numerical methods 65M15: Error bounds

Keywords
embedded waveform relaxation partial functional differential equations fixed and distributed delays superlinear convergence windowing technique

Citation

Liu, Jun; Jiang, Yao-Lin; Xu, Hong-Kun. Embedded Waveform Relaxation Methods for Parabolic Partial Functional Differential Equations. Taiwanese J. Math. 15 (2011), no. 2, 829--855. doi:10.11650/twjm/1500406237. https://projecteuclid.org/euclid.twjm/1500406237


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