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.
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