Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSOur final example is based upon the one-dimensional diffusion equation forthe temperature φ of a system:∂φ∂t = φκ∂2 ∂x 2 . (27.90)If φ i,j stands for φ(x 0 + i∆x, t 0 + j∆t) then a forward difference representationof the time derivative and a central difference representation for the spatialderivative lead to the following relationship:φ i,j+1 − φ i,j∆t= κ φ i+1,j − 2φ i,j + φ i−1,j(∆x) 2 . (27.91)This allows the construction of an explicit scheme for generating the temperaturedistribution at later times, given that it is known at some earlier time:φ i,j+1 = α(φ i+1,j + φ i−1,j )+(1− 2α)φ i,j , (27.92)where α = κ∆t/(∆x) 2 .Although this scheme is explicit, it is not a good one because of the asymmetricway in which the differences are formed. However, the effect of this can beminimised if we study and correct for the errors introduced in the following way.Taylor’s series for the time variable givesφ i,j+1 = φ i,j +∆t ∂φ i,j+ (∆t)2 ∂ 2 φ i,j∂t 2! ∂t 2 + ··· , (27.93)using the same notation as previously. Thus the first correction term to the LHSof (27.91) is− ∆t ∂ 2 φ i,j2 ∂t 2 . (27.94)The first term omitted on the RHS of the same equation is, by a similar argument,−κ 2(∆x)2 ∂ 4 φ i,j4! ∂x 4 . (27.95)But, using the fact that φ satisfies (27.90), we obtain∂ 2 φ∂t 2= ∂ ∂t( ) ( )κ ∂2 φ∂x 2 = κ ∂2 ∂φ∂x 2 = κ 2 ∂4 φ∂t ∂x 4 , (27.96)and so, to this accuracy, the two errors (27.94) and (27.95) can be made to cancelif α is chosen such that− κ2 ∆t2= − 2κ(∆x)2 , i.e. α = 1 4!6 .1032