This preview shows page 1. Sign up to view the full content.
Unformatted text preview: h. As illustrated in Figure 7.6.4(b) for n = 4, label the grid points using a rowwise
ordering scheme—i.e., label them as you would label matrix entries. u(x, y ) = g (x, y ) on the boundary O
C ∇ u = 0 in the interior
2 u(x, y ) = g (x, y ) on the boundary u(x, y ) = g (x, y ) on the boundary Y
P u(x, y ) = g (x, y ) on the boundary 00 01 02 03 04 05 10 11 12 13 14 15 20 21 22 23 24 25 30 31 32 33 34 35 40 41 42 43 44 45 51 52 53 54 55 h
h (a) (b)
Figure 7.6.4 76 Johann Peter Gustav Lejeune Dirichlet (1805–1859) held the chair at G¨ttingen previously
occupied by Gauss. Because of his work on the convergence of trigonometric series, Dirichlet
is generally considered to be the founder of the theory of Fourier series, but much of the
groundwork was laid by S. D. Poisson (p. 572) who was Dirichlet’s Ph.D. advisor. Copyright c 2000 SIAM Buy online from SIAM
http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com
Eigenvalues and Eigenvectors
Approximate ∂ 2 u/∂x2 and ∂ 2 u/∂y 2 at the interior grid points (xi , yj ) by using
the second-order centered diﬀerence formula (1.4.3) developed on p. 19 to write
∂x2 It is illegal to print, duplicate, or distribute this material
Please report violations to firstname.lastname@example.org ∂2u
∂y 2 (xi ,yj ) (xi ,yj ) = u(xi − h, yj ) − 2u(xi , yj ) + u(xi + h, yj )
+ O(h2 ),
h2 u(xi , yj − h) − 2u(xi , yj ) + u(xi , yj + h)
+ O(h2 ).
h2 (7.6.6) Adopt the notation uij = u(xi , yj ), and add the expressions in (7.6.6) using
∇2 u|(xi ,yj ) = 0 for interior points (xi , yj ) to produce
4uij = (ui−1,j + ui+1,j + ui,j −1 + ui,j +1 ) + O(h4 ) for D
E i, j = 1, 2, . . . , n. T
H In other words, the steady-state temperature at an interior grid point is approximately the average of the steady-state temperatures at the four neighboring grid
points as illustrated in Figure 7.6.5.
i − 1, j i, j − 1 ij IG
R uij = i, j + 1 ui−1,j + ui+1,j + ui,j −1...
View Full Document