Domain decomposition for parallelization of finite difference schemes for parabolic equations in wave propagation

Radio Science ◽  
1996 ◽  
Vol 31 (4) ◽  
pp. 943-954
Author(s):  
Sherman W. Marcus ◽  
David Degani
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Domain Decomposition ◽  
Parabolic Equations ◽  
Difference Schemes ◽  
Finite Difference Schemes

Stability of Finite-difference Schemes for Semilinear Multidimensional Parabolic Equations

2012 ◽  
Vol 12 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Bosko Jovanovic ◽  
Magdalena Lapinska-Chrzczonowicz ◽  
Aleh Matus ◽  
Piotr Matus
Keyword(s):  
Finite Difference ◽  
Parabolic Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Nonlinear Odes ◽  
The Stability ◽  
Initial Boundary Value

Abstract Abstract — We have studied the stability of finite-difference schemes approximating initial-boundary value problem (IBVP) for multidimensional parabolic equations with a nonlinear source of a power type. We have obtained simple sufficient input data conditions, in which the solutions of differential and difference problems are globally bounded for all t. It is shown that if these conditions are not satisfied, then the solution can blow-up (go to infinity) in finite time. The lower bound of the blow-up time has been determined. The stability of the difference solution has been proven. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear ODEs, Bihari-type inequalities and their discrete analogs.

PARALLEL DOMAIN DECOMPOSITION METHODS WITH THE OVERLAPPING OF SUBDOMAINS FOR PARABOLIC PROBLEMS

1996 ◽  
Vol 06 (08) ◽  
pp. 1169-1185 ◽  
Author(s):  
GRIGORII I. SHISHKIN ◽  
PETR N. VABISHCHEVICH
Keyword(s):  
Finite Difference ◽  
Domain Decomposition ◽  
Domain Decomposition Method ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
Difference Schemes ◽  
Parabolic Problems ◽  
Uniform Grid ◽  
Finite Difference Schemes ◽  
Dimensional Boundary

For a model of two-dimensional boundary value problem for a second-order parabolic equation, finite difference schemes on the base of a domain decomposition method, oriented on modern parallel computers, is constructed. In the used finite difference schemes iterations at time levels are not applied; some subdomains overlap. We study two classes of schemes characterized by synchronous and asynchronous implementations. It is shown that, under refining grids, the approximate solutions do converge to the exact one in the uniform grid norm.

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