Download PDF by Franz J. Vesely (auth.): Computational Physics: An Introduction

By Franz J. Vesely (auth.)

ISBN-10: 1475723075

ISBN-13: 9781475723076

ISBN-10: 1475723091

ISBN-13: 9781475723090

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1_- b·-"'·h· f3i + /i 9i -ai =--f3i /i 9i + (i=N-1, ... 32 for the starting value x2 = 91x1 +ht-) EXAMPLE: x1 i = 1, ... 'N -1 follows from (31 x 1 In A· x = b, let 1 0 3 1 1 4 0 1 Downward recursion (Equ. 33) + 1 1 x 2 = b1 and 32 Chapter 2 i = 3: Linear algebra 1 h2=10 h - 19 1 - 27 3 g2= - - 10 20 i = 2: g1 = - 27 Upward recursion (Equ. 33): 8 34 9 X1 i = 1: X2 i = 2: X3 i = 3: X4 17 1 17 = 23 17 A similar method which may be used in the case of a five-diagonal matrix is given in [ENGELN 91].

The ADI scheme is tailored to the numerical treatment of the potential equation 'V 2 u = -p. 1. For the time being, suffice it to say that the method converges even more rapidly than SOR acccelerated ala Chebysheff. ; h 1 points out the direction of the gradient conjugate to g 0 . The steepest descent method follows the tedious zig-zag course Po --t P1 --t P2 --t . . The conjugate gradient h 1 gets us to the goal in just two steps. 5 Conjugate Gradient Method ( CG) The task of solving the equation A · x = b may be interpreted as a minimization problem.

36) forgoing the ambition to reach the correct answer in one single step. 38) This procedure can be shown to converge to the solution of A· x = b if, and only if, lxk+l- Xk I < lxk- Xk-11· This, however, is the case if all eigenvalues of the matrix B- 1 . [B- A] are situated within the unit circle. It is the choice of the matrix B where the various iterative methods differ. The three most important methods are known as Jacobi relaxation, GaussSeidel relaxation (GSR) and successive over-relaxation (SOR).

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