Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation Galerkin Method

2006 ◽  
Author(s):  
James Geer ◽  
John Fillo
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Galerkin Method ◽  
Difference Equations ◽  
Space Variable ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hybrid Technique ◽  
Finite Difference Equations ◽  
Hybrid Equations

A new technique for the development of finite difference schemes for diffusion equations is presented. The model equations are the one space variable advection diffusion equation and the two space variable diffusion equation, each with Dirichlet boundary conditions. A two-step hybrid technique, which combines perturbation methods, based on the parameter ρ = Δt / (Δx)2, with the Galerkin method, provides a systematic way to develop new finite difference methods, referred to as hybrid equations. The main contributions of this paper include: 1) recovery of classical explicit or implicit finite difference schemes using only the perturbation terms; 2) development of new finite difference schemes, referred to as hybrid equations, which have better stability properties than the classical finite difference equations, permitting the use of larger values of the parameter ρ; and 3) higher order accurate methods, with either O((Δx)4) or O((Δx)6) truncation error, formed by convex linear combinations of the classical and hybrid equations. The solution of the hybrid finite difference equations requires only a tridiagonal equation solver and, hence, does not lead to excessive computational effort.

Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation–Galerkin Method

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
James Geer ◽  
John Fillo
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Galerkin Method ◽  
Difference Equations ◽  
Space Variable ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hybrid Technique ◽  
Finite Difference Equations ◽  
Hybrid Equations

A new technique for the development of finite difference schemes for diffusion equations is presented. The model equations are the one space variable advection diffusion equation and the two space variable diffusion equation, each with Dirichlet boundary conditions. A two-step hybrid technique, which combines perturbation methods based on the parameter ρ=Δt∕(Δx)2 with the Galerkin method, provides a systematic way to develop new finite difference methods, referred to as hybrid equations. The main contributions of this paper include the (1) recovery of classical explicit or implicit finite difference schemes using only the perturbation terms; (2) development of new finite difference schemes, referred to as hybrid equations, which have better stability properties than the classical finite difference equations, permitting the use of larger values of the parameter ρ; and (3) higher order accurate methods, with either O((Δx)4) or O((Δx)6) truncation error, formed by convex linear combinations of the classical and hybrid equations. The solution of the hybrid finite difference equations requires only a tridiagonal equation solver and, hence, does not lead to excessive computational effort.

A hybrid technique combining the method of moments in the time domain and FDTD

1998 ◽  
Vol 8 (8) ◽  
pp. 281-283 ◽  
Author(s):  
A.R. Bretones ◽  
R. Mittra ◽  
R.G. Martin
Keyword(s):  
Method Of Moments ◽  
Time Domain ◽  
Hybrid Technique ◽  
The Time Domain
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