Order the unknowns by rows but gather all of the

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Unformatted text preview: venience as 2 32 32 3 C1 D1 x1 b1 6D C D 7 6 x2 7 6 b2 7 76 Ax = 6 2 2 . . 2 6 76 . 7 = 6 . 7 4 54 . 7 6 . 7 . .54.5 DK ;1 CK ;1 xK ;1 bK ;1 The matrices Ck, etc. were de ned by (8.1.4c-8.1.4d). The SOR procedure is applied to an entire row, i.e., we compute ) ^ Ck x(k +1) = ;Dk x(k;+11) ; Dk x(k+1 + bk ^ x(k +1) = !x(k +1) + (1 ; !)x(k ) k = 1 2 ::: K ; 1 = 0 1 ::: : Thus, we have to solve a tridiagonal system at each step of the process. This p procedure converges faster than point SOR iteration by a factor of 2. 4. Alternating direction implicit (ADI) methods. By considering an elliptic problem as the steady state limit of a transient parabolic problem, we can use some methods for time-dependent problems to solve them. In particular, the ADI method (cf. Section 5.2) has been adapted to the solution of elliptic problems. The goal, when using this approach, is to select the \time step" so that the ADI scheme converges to steady state as fast as possible. The ADI method with a single acceleration parameter (arti cial time step) has the same convergence rate as SOR. Further acceleration is possible by choosing a sequence of arti cial time steps, changing them after each predictor-corrector sweep, and applying them cyclically. Wachspress 7] has shown how to select nearly optimal acceleration parameters. The results shown in Table 9.2.5 summarize the convergence rates of the methods that we have studied in this section. They are all obtained by solving a Dirichlet problem on a 9.2. Basic Iterative Solution Methods 29 square mesh with uniform spacing h = x = y. The convergence rates of all methods decline as the mesh spacing decreases. Degradation in performance is least with the ADI method however, computing optimal parameters can be problematical in realistic situations. Method Conv. Rate Jacobi h2 Gauss-Seidel 2h2 SOR (with !OPT ) 2h 8 ADI ( with m parameters) m ( h )1=m 2 Table 9.2.5: Convergence rates for various iterative methods as a function of mesh spacing h for a Dirichlet problem on...
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