Download e-book for iPad: Computational Physics: An Introduction by Franz J. Vesely (auth.)

By Franz J. Vesely (auth.)

ISBN-10: 1475723075

ISBN-13: 9781475723076

ISBN-10: 1475723091

ISBN-13: 9781475723090

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Further applications of linear algebra will be treated in Chapter 3 (Stochastics). 4 we have shown how to discretize the diffusion equation (or equation of thermal conduction) by applying the DNGF and DDST formulae. 85) In this manner we arrived at the "FTCS-"formula. x) 2 , - au~+l ~-1 + (1 + 2a)u~+l ~ au~+l ~+1 = u~ ~ for i = 1, ... N- 1. Once more fixing the boundary values u 0 and may write this system of equations in matrix form, thus: A. 90) 48 Chapter 2 Linear algebra It is now an easy matter to invert this tridiagonal system by the recursion scheme of Sec.

A sequence of random numbers Xn of type real, equidistributed over the interval (0, 1 ), may he obtained by dividing In by m. Most of the FORTRAN versions offered by the various manufacturers and most of the other high-level languages contain some internal routine based on this technique. These routines are usually called by names like RAND, RND, RAN etc. ) The first number in the sequence, the - odd-numbered - "seed" 10 , may often be chosen by the user. 2 In fact, there have been many attempts to construct "physical" random number generators which may be based on thermal noise in resistors or on quantum phenomena [STAUFFER 89).

921 ) . 003 X2- . 54 is "optimal" only in the asymptotic sense, that is, after a certain number of iterations. During the first few iterative steps the SOR procedure may give rise to overshooting corrections - particularly if w is distinctly larger than 1. 54. This procedure, which is known as "Chebysheff acceleration", consists of the following steps: 38 Chapter 2 Linear algebra • The solution vector x is split in 2 vectors Xe, X 0 consisting of the elements Xi with even and odd indices, respectively; the vector b is split up in the same manner.

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

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