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 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 ; and (3) higher order accurate methods, with either or 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.
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Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation–Galerkin Method
James Geer,
James Geer
Professor Emeritus
Watson School of Engineering and Applied Science,
Binghamton University
, Binghamton, NY 13902
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John Fillo
John Fillo
Associate Dean for Research and External Affairs
Watson School of Engineering and Applied Science,
Binghamton University
, Binghamton, NY 13902
Search for other works by this author on:
James Geer
Professor Emeritus
Watson School of Engineering and Applied Science,
Binghamton University
, Binghamton, NY 13902
John Fillo
Associate Dean for Research and External Affairs
Watson School of Engineering and Applied Science,
Binghamton University
, Binghamton, NY 13902J. Heat Transfer. Jun 2008, 130(6): 061701 (10 pages)
Published Online: April 21, 2008
Article history
Received:
March 14, 2006
Revised:
January 30, 2008
Published:
April 21, 2008
Citation
Geer, J., and Fillo, J. (April 21, 2008). "Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation–Galerkin Method." ASME. J. Heat Transfer. June 2008; 130(6): 061701. https://doi.org/10.1115/1.2891135
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