A Practical Guide to Pseudospectral Methods by Bengt Fornberg

By Bengt Fornberg

Prior to now twenty years, pseudospectral tools have emerged as profitable, and sometimes more suitable, choices to raised recognized computational strategies, reminiscent of finite distinction and finite point tools of numerical answer, in numerous key program parts. those components contain computational fluid dynamics, wave movement, and climate forecasting. This ebook explains how, while and why this pseudospectral process works. that allows you to make the topic available to scholars in addition to researchers and engineers, the writer offers the topic utilizing illustrations, examples, heuristic motives, and algorithms instead of rigorous theoretical arguments. This ebook may be of curiosity to graduate scholars, scientists, and engineers drawn to employing pseudospectral how to genuine difficulties.

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FORTRAN codes (and a test driver) are given in Appendix C. The derivation of the algorithm starts by noting that the best polynomialbased approximation to d"f/dx k at some point ~ is obtained by differentiating Lagrange's interpolating polynomial (to be described shortly; cf. 3-1»: Thus I -- ~ dkFj(x) I dkpN(X) dx x=~ --k- ~ j=O k dx x=~ · f(x· ). 1-2) Considering not only interpolations based on all gridpoints Xj' j = 0, I, ... , N, but also interpolations based on leading subsets (xj ' j = 0, I, ...

I ·(X) I,) F i i(X) , Xj-Xi 1- ,) , = rr~:~(Xi-1 . I -X,) (X-Xi-I)Fi - 1, i-I(X). 1-3) into these relations gives, after equating coefficients, two recursions for the weights c{j. o = 1, these recursions suffice to determine all Ci~j' k = 0,1, ... , m, j = k, k+ 1, ... , N, in a fast and numerically stable manner. 2. 2-2 illustrate how the magnitudes of the weights for the first derivative grow with increasing orders of accuracy (cf. 1-2). In the centered case, approximations of increasing orders of accuracy converge to a limit method of forplally infinite order.

However, the proportionality constant is much higher than for FFTs, so the approach appears not to be competitive in the present context. , 21 gridpoints if both ends are included, 20 within the period for periodic problems). Part (a) shows the (periodic) stencil [1 -2 1]/h 2 and part (b) the periodic PS matrix. 3-1. (c) PS, nonperiodic, equi-spaced grid [13 x 10 6 ]. (d) PS, non periodic, Chebyshev [17 x 10 3 ]. 50 4. 5, respectively). Large elements are seen in the top and bottom rows (corresponding to approximations near the boundaries).

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