A Practical Guide to Pseudospectral Methods by Bengt Fornberg

By Bengt Fornberg

In the past twenty years, pseudospectral tools have emerged as profitable, and infrequently enhanced, possible choices to raised identified computational strategies, similar to finite distinction and finite point equipment of numerical resolution, in different key software parts. those parts comprise computational fluid dynamics, wave movement, and climate forecasting. This ebook explains how, while and why this pseudospectral method works. so one can make the topic available to scholars in addition to researchers and engineers, the writer offers the topic utilizing illustrations, examples, heuristic factors, and algorithms instead of rigorous theoretical arguments. This e-book can be of curiosity to graduate scholars, scientists, and engineers attracted to using pseudospectral tips on how to genuine difficulties.

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FD4 FD4 ... ,,-. ,_ .... -. ,...... 1I-'iI!. fie. •... :''. ~"'~,. ' ! \/"' . ' . '. r. \~/' , ••,•• ~ .. ~ . "' .... "'. 2-3. Errors when approximating the derivative of a step function: local approximation to the derivative near a step versus the long-term evolution of numerical solutions with a step initial condition. U(X,O) = I-H(x) = [ I ° ° if x < 0, Of I X~ , the analytical solution to the new equation becomes 1 1 U(X,t)=--- 2 At x 21r = 0, its x derivative is au I = __1_ ax x=o l l OO -00 1 -[(coschpwP+1t)(sinwx) w.

2) Only four operations are needed for each weight (to leading order; note that the subtractions Xi - ~ and Xi - Xj can be moved out of the innermost loop). (3) The calculation of weights is numerically stable. However, especially in case of high derivatives, applying FD weights to a function can be numerically ill-conditioned and can lead to severe cancellations and loss of significant digits. For analytic functions that can be evaluated in the complex plane, finite difference formulas can be very accurate and quite well-conditioned also for high derivatives.

Noting the identity Lk=_oo[(-l)k/(k+x)] = 1r/(sin 1rx), we have 3. 5-1) if i = j. 6. Equivalence of PS methods and limits of FD methods Periodic case. 5-1» are identical; hence, the two methods are equivalent. 3), and so forth. For details, see Fornberg (1990a). Nonperiodic case. We assume that the data cannot be extended past the boundaries. The order of accuracy for the approximations correspond to the number of gridpoints (rather than being formally infinite). The PS method now turns out to be equivalent to using the FD approximations, whose stencils extend over all the gridpoints.

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