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

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.