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446 chapter 17A less obvious improvement in the relaxation technique, known as successiveoverrelaxation (SOR), starts by writing the algorithm (17.25) in a form thatdetermines the new values of the potential U (new) as the old values U (old) plusa correction or residual r:U (new)i,j= U (old)i,j + r i,j . (17.29)While the Gauss–Seidel technique may still be used to incorporate the updatedvalues in U (old) to determine r, we rewrite the algorithm here in the general form:r i,j ≡ U (new)i,j − U (old)i,j= 1 []U (old)i+1,j4+ U (new)i−1,j + U (old)i,j+1 + U (new)i,j−1 − U (old)i,j . (17.30)The successive overrelaxation technique [Pres 94, Gar 00] proposes that if convergenceis obtained by adding r to U, then even more rapid convergence might beobtained by adding more or less of r:U (new)i,j= U (old)i,j + ωr i,j , (SOR), (17.31)where ω is a parameter that amplifies or reduces the residual. The nonacceleratedrelaxation algorithm (17.28) is obtained with ω =1, accelerated convergence (overrelaxation)is obtained with ω ≥ 1, and underrelaxation is obtained with ω2 may lead to numerical instabilities.Although a detailed analysis of the algorithm is needed to predict the optimal valuefor ω, we suggest that you explore different values for ω to see which one worksbest for your particular problem.17.4.2 Lattice PDE ImplementationIn Listing 17.1 we present the code LaplaceLine.java that solves the square-wireproblem (Figure 17.1). Here we have kept the code simple by setting the length ofthe box L = N max ∆ = 100 and by taking ∆=1:U(i, N max ) = 99 (top), U(1,j)=0 (left),U(N max ,j)= 0 (right), U(i, 1) = 0 (bottom),(17.32)We run the algorithm (17.27) for a fixed 1000 iterations. A better code would vary ∆and the dimensions and would quit iterating once the solution converges to sometolerance. Study, compile, and execute the basic code.−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 446