Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODS27.25 Laplace’s equation,∂ 2 V∂x + ∂2 V2 ∂y =0, 2is to be solved for the region and boundary conditions shown in figure 27.7.V =80−∞40404040404040∞202020V =0Figure 27.7Region, boundary values and initial guessed solution values.Starting from the given initial guess for the potential values V , and using thesimplest possible form of relaxation, obtain a better approximation to the actualsolution. Do not aim to be more accurate than ± 0.5 units, and so terminate theprocess 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 arelaxation method on a square grid with common spacing h. Asinthemaintext,denote φ(x 0 + ih, y 0 + jh) byφ i,j . Further, define φ m,ni,jbyφ m,ni,j≡ ∂m+n φ∂x m ∂y nevaluated at (x 0 + ih, y 0 + jh).(a) Show thatφ 4,0i,j+2φ 2,2i,j+ φ 0,4i,j=0.(b) Working up to terms of order h 5 , find Taylor series expansions, expressed interms of the φ m,ni,j ,for S ±,0 = φ i+1,j + φ i−1,j ,S 0,± = φ i,j+1 + φ i,j−1 .(c) Find a corresponding expansion, to the same order of accuracy, for φ i±1,j+1 +φ i±1,j−1 and hence show thatS ±,± = φ i+1,j+1 + φ i+1,j−1 + φ i−1,j+1 + φ i−1,j−1has the form)+h44φ 0,0i,j+2h 2 (φ 2,0i,j+ φ 0,2i,j6 (φ4,0 i,j+6φ 2,2i,j+ φ 0,4i,j ).(d) Evaluate the expression 4(S ±,0 +S 0,± )+S ±,± and hence deduce that a possiblerelaxation scheme, good to the fifth order in h, is to recalculate each φ i,j asthe weighted mean of the current values of its four nearest neighbours (eachwith weight 1 1) and its four next-nearest neighbours (each with weight ).5 201038