Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.8 PARTIAL DIFFERENTIAL EQUATIONS

than an interval divided into equal steps by the points at which solutions to the

equations are to be found, a mesh of points in two or more dimensions has to be

set up and all the variables given an increased number of subscripts.

Considerations of the stability, accuracy and feasibility of particular calcula-

tional schemes are the same as for the one-dimensional case in principle, but in

practice are too complicated to be discussed here.

Rather than note generalities that we are unable to pursue in any quantitative

way, we will conclude this chapter by indicating in outline how two familiar

partial differential equations of physical science can be set up for numerical

solution. The first of these is Laplace’s equation in two dimensions,

∂^2 φ

∂x^2

+

∂^2 φ

∂y^2

=0, (27.86)

the value ofφbeing given on the perimeter of a closed domain.

A grid with spacings ∆xand ∆yin the two directions is first chosen, so that,

for example,xistands for the pointx 0 +i∆xandφi,jfor the valueφ(xi,yj). Next,

using a second central difference formula, (27.86) is turned into

φi+1,j− 2 φi,j +φi− 1 ,j

(∆x)^2

+

φi,j+1− 2 φi,j +φi,j− 1

(∆y)^2

=0,

(27.87)

fori=0, 1 ,...,Nandj=0, 1 ,...,M.If(∆x)^2 =λ(∆y)^2 then this becomes the

recurrence relationship

φi+1,j+φi− 1 ,j+λ(φi,j+1+φi,j− 1 )=2(1+λ)φi,j. (27.88)

The boundary conditions in their simplest form (i.e. for a rectangular domain)

mean that

φ 0 ,j,φN,j,φi, 0 ,φi,M (27.89)

have predetermined values. Non-rectangular boundaries can be accommodated,

either by more complex boundary-value prescriptions or by using non-Cartesian

coordinates.

To find a set of values satisfying (27.88), an initial guess of a complete set of

values for theφi,jis made, subject to the requirement that the quantities listed

in (27.89) have the given fixed values; those values that are not on the boundary

are then adjusted iteratively in order to try to bring about condition (27.88)

everywhere. Clearly one scheme is to setλ= 1 and recalculate eachφi,jas the

mean of the four current values at neighbouring grid-points, using (27.88) directly,

and then to iterate this recalculation until no value ofφchanges significantly

after a complete cycle through all values ofiandj. This procedure is the simplest

of such ‘relaxation’ methods; for a slightly more sophisticated scheme see exercise

27.26 at the end of this chapter. The reader is referred to specialist books for

fuller accounts of how this approach can be made faster and more accurate.