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pdes for electrostatics & heat flow 443xyi-1, ji, j-1i, j i+1, ji, j+1Figure 17.3 The algorithm for Laplace’s equation in which the potential at the point(x, y)=(i, j)∆ equals the average of the potential values at the four nearest-neighbor points.The nodes with white centers correspond to fixed values of the potential along theboundaries.than fall off abruptly, the series develops large oscillations that tend to overshootthe function at the corner. To obtain a smooth solution, we had to sum 40,000 terms,where, in contrast, the numerical solution required only hundreds of iterations.17.4 Solution: Finite-Difference MethodTo solve our 2-D PDE numerically, we divide space up into a lattice (Figure 17.3)and solve for U at each site on the lattice. Since we will express derivatives interms of the finite differences in the values of U at the lattice sites, this is calleda finite-difference method. A numerically more efficient, but also more complicatedapproach, is the finite-element method (Unit II), which solves the PDE for smallgeometric elements and then matches the elements.To derive the finite-difference algorithm for the numeric solution of (17.5), wefollow the same path taken in § 7.1 to derive the forward-difference algorithm fordifferentiation. We start by adding the two Taylor expansions of the potential tothe right and left of (x, y) and above and below (x, y):U(x +∆x, y)=U(x, y)+ ∂U∂x ∆x + 1 ∂ 2 U2 ∂x 2 (∆x)2 + ··· , (17.20)U(x − ∆x, y)=U(x, y) − ∂U∂x ∆x + 1 ∂ 2 U2 ∂x 2 (∆x)2 −··· . (17.21)−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 443