Download e-book for kindle: A Practical Guide to Pseudospectral Methods by Bengt Fornberg

By Bengt Fornberg

ISBN-10: 0521495822

ISBN-13: 9780521495820

Prior to now 20 years, pseudospectral tools have emerged as winning, and infrequently improved, possible choices to higher recognized computational approaches, comparable to finite distinction and finite aspect equipment of numerical answer, in different key program parts. those components comprise computational fluid dynamics, wave movement, and climate forecasting. This booklet explains how, whilst and why this pseudospectral procedure works. as a way to make the topic available to scholars in addition to researchers and engineers, the writer offers the topic utilizing illustrations, examples, heuristic causes, and algorithms instead of rigorous theoretical arguments. This publication could be of curiosity to graduate scholars, scientists, and engineers attracted to using pseudospectral how to genuine difficulties.

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1-1. Functionsf~,1J(x) = 1/(1 + [(X-~)/f1l2) with minimal 11 (Le. maximal curvature at the tip). (a) Equi-spaced nonperiodic 'PS: borderline convergence/ divergence as N -+ 00. 9. 4-1). 1-1(a) shows these most peaked functions f~, l1(X) for different values of ~ E [-1, 1]. The equi-spaced PS method is clearly much better able to resolve high curvatures near the ends of [-1, 1] than in the interior. 1-1(a). 1. Convergence of PS methods for smooth functions 39 This issue is discussed also by Solomonoff and Turkel (1989).

2-1 shows the time evolution for u(x, t); the one for v(x, t) is qualitatively similar. After the pulses have hit the interfaces at x = 0 and x = ±1 numerous times - on each occasion generating two outgoing pulses (transmitted and reflected) - the analytic solution at t = 7 consists of just three pulses. 2-2. We have used grids where 0 and ±1 fall halfway between gridpoints, thus saving us from deciding on the values of c(x) at these locations. No numerical smoothing was used in any of these cases.

One approach for controlling such errors is to apply weak damping (dissipation). However, as we shall see, anyone who views aliasing solely as a source of errors is missing out on one of the most important (and intriguing) strengths of PS methods. 2. Convergence of PS methods for nonsmooth functions The PS method often performs extremely well even in cases where both solutions and variable coefficients are not smooth. ). Example. 2-1) with a sharp pulse as the initial condition. 2-1) supports solutions of the following forms, traveling to the right (=» and to the left (¢::): in [-1,0], {U(X, t) = -v(x, t) =>, u(x, t) = +v(x, t) ¢::; .

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A Practical Guide to Pseudospectral Methods by Bengt Fornberg

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