Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

27.25 Laplace’s equation,

∂^2 V

∂x^2

+

∂^2 V

∂y^2

=0,

is to be solved for the region and boundary conditions shown in figure 27.7.

40 40 40 40 40 40 40

20 20 20

V=80

V=0

−∞ ∞

Figure 27.7 Region, boundary values and initial guessed solution values.

Starting from the given initial guess for the potential valuesV, and using the
simplest possible form of relaxation, obtain a better approximation to the actual
solution. Do not aim to be more accurate than±0.5 units, and so terminate the
process when subsequent changes would be no greater than this.
27.26 Consider the solution,φ(x, y), of Laplace’s equation in two dimensions using a
relaxation method on a square grid with common spacingh.Asinthemaintext,
denoteφ(x 0 +ih, y 0 +jh)byφi,j. Further, defineφm,ni,j by

φm,ni,j≡

∂m+nφ

∂xm∂yn

evaluated at (x 0 +ih, y 0 +jh).

(a) Show that

φ^4 i,j,^0 +2φ^2 i,j,^2 +φ^0 i,j,^4 =0.

(b) Working up to terms of orderh^5 , find Taylor series expansions, expressed in

terms of theφm,ni,j,for

S±, 0 =φi+1,j+φi− 1 ,j,

S 0 ,±=φi,j+1+φi,j− 1.

(c) Find a corresponding expansion, to the same order of accuracy, forφi± 1 ,j+1+

φi± 1 ,j− 1 and hence show that

S±,±=φi+1,j+1+φi+1,j− 1 +φi− 1 ,j+1+φi− 1 ,j− 1

has the form

4 φ^0 i,j,^0 +2h^2 (φ^2 i,j,^0 +φ^0 i,j,^2 )+

h^4

6

(φi,j^4 ,^0 +6φ^2 i,j,^2 +φ^0 i,j,^4 ).

(d) Evaluate the expression 4(S±, 0 +S 0 ,±)+S±,±and hence deduce that a possible

relaxation scheme, good to the fifth order inh, is to recalculate eachφi,jas

the weighted mean of the current values of its four nearest neighbours (each

with weight^15 ) and its four next-nearest neighbours (each with weight 201 ).