Linearly Implicit Domain Decomposition Methods for Nonlinear Time-Dependent Reaction-Diffusion Problems

2009 ◽  
pp. 267-274
Author(s):  
A. Arrarás ◽  
L. Portero ◽  
J. C. Jorge
Keyword(s):  
Domain Decomposition ◽  
Reaction Diffusion ◽  
Decomposition Methods ◽  
Time Dependent ◽  
Domain Decomposition Methods ◽  
Diffusion Problems

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.

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