定常1次元拡散問題に対する-数値解法に関するノート

概要

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A numerical method is proposed for solving the steady one-dimensional diffusion problem described by Pe-1 y″y′-f (y) =0 under the boundary conditions given by Danckwerts, y′=Pe (y-yF) at x=0 and y′= 0 at x=1. In this method the central finite difference equation is used for calculating y″j and y′j (0≤ j ≤N-1), where j=0 and j=N denote x=0 and x=1, respectively. However, the feature of this method is to use the following finite difference equation, which gives the relationship between y N-1 and yN. yN-1-yN-Peh2/2Peh+4 {f (y N-1) +f (yN)} =0 This equation is derived by using y″N-1 = (Δy′N-1 + ∇y′N) /2 and y′N-1 = (ΔyN-1 + ∇ yN) /2 with the boundary condition y′N=0 where the symbols Δ and ∇ show the forward and backward difference operators. The proposed method provides a stable solution at large Pe number as illustrated in numerical examples.
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