By L. Collatz (auth.), J. G. Gram (eds.)
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Additional resources for Numerical Solution of Partial Differential Equations: Proceedings of the NATO Advanced Study Institute held at Kjeller, Norway, August 20–24, 1973
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Semi-iteration (alternative to 4a). 2) with a = 0, ~ = S(~l)' 5. Convergence tests. 6) lliill ~ < A If this test is satisfied t· The semi-iterative method is carried out until < 1:. 8 ) 9. -ull A~ NUMERICAL RESULTS We now describe some numerical experiments which were carried out using the accelerated SSOR method. 2) was solved in the unit square with boundary values zero on SOLUTION OF LINEAR SYSTEMS OF EQUATIONS 51 all sides except where y = O. On the side y = 0, the values of u(x,y) were unity.
2. Carre, B. A. "The determination of the optimum accelerating factor for successive over-relaxation," The Computer Journal 4, 73-78 (1961). 3. Concus, Paul, and Gene H. Golub. "Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations," Report STAN-CS-72-278, Computer Science Department, Stanford University, April 1972. 4. Golub, G. H. "The use of Chebyshev matrix polynomials in the iterative solution of linear systems compared with the method of successive overrelaxation," doctoral thesis, University of Illinois, Urbana, Illinois (1959).
3. BASIC ITERATIVE METHODS We now define three "basic" iterative procedures, namely, the Jacobi, Gauss-Seidel, and successive overrelaxation methods. 1) Au b. 3) B = 1/4 and c (1/4)b. 4 ) B = In general we let -1 I - D A where D is a diagonal matrix with the same diagonal elements as A. The Jacobi method is defined as follows. 5) u(n+l) = Bu(n) + c. 6) 1 (n) 1 (n) + cl + ~3 1 (n) 1 (n) + c2 + ~4 u l(n+l) = ~2 (n+l) u2 = ~1 etc. Note that even though ui n+l ) is available when we compute u(n+l) we use the old value ul(n) instead.