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.

Some recent applications of asymptotic error expansions to finite-difference schemes

1971 ◽  
Vol 323 (1553) ◽  
pp. 167-177 ◽  
Keyword(s):  
Finite Difference ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Linear Multistep Methods ◽  
Finite Difference Schemes ◽  
The Finite Element Method ◽  
Technical Device ◽  
Asymptotic Error ◽  
The Subject ◽  
Asymptotic Error Expansions

In this paper we will present some results on asymptotic error expansions of the approximate solutions of differential equations by finite-difference schemes. We will present some quite well-known material, in an effort to make the presentation self-contained, as well as discuss recent work by Engquist on linear multistep methods for initial-value problems, by Kreiss on extrapolation procedures for elliptic finite-difference schemes, by Pereyra and co-workers on iterated deferred-correction methods for elliptic equations and by the author and Hald on the finite-element method. Practical aspects of the subject as well as the use of error expansions as a technical device in theoretical numerical analysis are discussed.

scholarly journals Modified Finite Difference Schemes on Uniform Grids for Simulations of the Helmholtz Equation at Any Wave Number

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Hafiz Abdul Wajid ◽  
Naseer Ahmed ◽  
Hifza Iqbal ◽  
Muhammad Sarmad Arshad
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Bloch Wave ◽  
Difference Schemes ◽  
Wave Number ◽  
Uniform Grid ◽  
Finite Difference Schemes ◽  
Radiation Boundary Conditions ◽  
Fine Mesh ◽  
Uniform Grids

We construct modified forward, backward, and central finite difference schemes, specifically for the Helmholtz equation, by using the Bloch wave property. All of these modified finite difference approximations provide exact solutions at the nodes of the uniform grid for the second derivative present in the Helmholtz equation and the first derivative in the radiation boundary conditions for wave propagation. The most important feature of the modified schemes is that they work for large as well as low wave numbers, without the common requirement of a very fine mesh size. The superiority of the modified finite difference schemes is illustrated with the help of numerical examples by making a comparison with standard finite difference schemes.

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