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Research Papers: Forced Convection

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

[+] Author and Article Information
James Geer, John Fillo

Watson School of Engineering and Applied Science, Binghamton University, Binghamton, NY 13902

J. Heat Transfer 130(6), 061701 (Apr 21, 2008) (10 pages) doi:10.1115/1.2891135 History: Received March 14, 2006; Revised January 30, 2008; Published April 21, 2008

## Abstract

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.

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## Figures

Figure 7

Scaled relative error at x=y=0.5 plotted as a function of time for test problem 4 using various numerical schemes, with Δx=0.05 and ρ=1∕4. Note that the error of the hybrid method (Sec. 7) changes from O((Δx)4) to O((Δx)2) due to the nonhomogeneous boundary conditions.

Figure 1

Critical values ρ* of ρ plotted as a functionf of ξ for several values of Δ=VΔx∕2 for the explicit scheme (Eq. 20) for the first model problem. As the plot indicates, the scheme is (somewhat) more stable for negative values of ξ than for positive values of ξ.

Figure 2

Critical values ρ* of ρ plotted as a function of V for Δx=0.05 for the composite scheme (Eq. 40) for the first model problem. These critical values are about 75% larger than the corresponding values for the explicit scheme (see Fig. 1).

Figure 3

Scaled relative error at x=0.5 plotted as a function of time for test problem 1 using various numerical schemes, with Δx=0.1 and ρ=1∕3. Note that the O((Δx)4) error of the composite scheme (Eq. 32) and the O((Δx)6) error of the supercomposite scheme (Eq. 36) are clearly illustrated.

Figure 5

Scaled relative error at x=0.5 plotted as a function of time for test problem 2 using various numerical schemes, with Δx=0.1, ρ=1∕3, and V=1. Note that the O((Δx)4) error of the composite scheme (Eq. 40) is clearly illustrated.

Figure 6

Scaled relative error at x=y=0.5 plotted as a function of time for test problem 3 using various numerical schemes, with Δx=0.05 and ρ=1∕4. Note that the O((Δx)4) error of the hybrid method (Sec. 7) is clearly illustrated.

Figure 4

Scaled relative error at x=0.5 plotted as a function of time for test problem 1 using various numerical schemes, with Δx=0.1 and ρ=2∕3. Note that the O((Δx)4) error of the composite scheme (Eq. 32) and the O((Δx)6) error of the supercomposite scheme (Eq. 36) are clearly illustrated. Also, the explicit method is not used here since ρ>ρ*=1∕2 for this method.

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