CHAPTER TREFETHEN Chapter Fourier sp ectral metho ds An example Unb ounded grids Perio dic grids Stability Notes and references Think globally Act lo cally BUMPER STICKER CHAPTER TREFETHEN Finite dierence metho ds like nite element metho ds are based on lo cal representations of functionsusually by loworder p olynomials In contrast sp ectral metho ds make use of global representations usually by highorder p olynomials or Fourier series Under fortunate circumstances the result is a degree of accuracy that lo cal metho ds cannot match For largescale compu tations esp ecially in several space dimensions this higher accuracy may be most imp ortant for p ermitting a coarser mesh hence a smaller numb er of data values to store and op erate up on It also leads to discrete mo dels with little or no articial dissipation a particularly v aluable feature in high Reynolds number uid ow calculations where the small amountofphysical dissipation maybeeasilyoverwhelmed byany dissipation of the numerical kind Sp ectral metho ds have achieved dramatic successes in this area Some of the ideas b ehind sp ectral metho ds have been intro duced several times into numerical analysis One early prop onentwas Cornelius Lanczos in the s who showed the p ower of Fourier series and Chebyshev p olynomials in a variety of problems where they had not b een used b efore The emphasis was on ordinary dierential equations Lanczoss work has been carried on esp ecially in Great Britain byanumb er of colleagues suchasCWClenshaw More recently sp ectral metho ds were intro duced again by Kreiss and Oliger Orszag and others in the s for the purp ose of solving the par tial dierential equations of uid mechanics Increasingly they are becoming viewed within some elds as an equal comp etitor to the better established nite dierence and nite element approaches At present however they are less well understo o d Sp ectral metho ds fall into various categories and one distinction often made is between Galerkin tau and collo cation or pseudosp ectral sp ectral metho ds In a word the rst twowork with the co ecients of a global expansion and the latter with its values at p oints The discussion in this b o ok is entirely conned to collo cation metho ds which are probably used the most often chiey b ecause they oer the simplest treatment of nonlinear terms Sp ectral metho ds are aected far more than nite dierence metho ds by the presence of b oundaries which tend to intro duce stability problems that are illundersto o d and sometimes highly restrictive as regards time step Indeed diculties with b oundaries direct and indirect are probably the primary rea son why sp ectral metho ds have not replaced their loweraccuracy comp etition CHAPTER TREFETHEN in most applications Chapter considers sp ectral metho ds with b oundaries but the presentchapter assumes that there are none This means that the spa tial domain is either innitea theoretical device not applicable in practice or p erio dic In those cases where the physical problem naturally inhabits a p erio dic domain sp ectral metho ds maybe strikingly successful Conspicuous examples are the global circulation mo dels used by meteorologists Limited area meteorological co des since they require b oundaries are often based on nite dierence formulas but as of this writing almost all of the global circula tion co des in use which mo del ow in the atmosphere of the entire spherical earth are sp ectral AN EXAMPLE TREFETHEN An example Sp ectral metho ds have b een most dramatically successful in problems with p erio dic geometries In this section we presenttwo examples of this kind that involve elastic wave propagation Both are taken from B Fornb erg The pseudosp ectral metho d comparisons with nite dierences for the elastic wave equation Geophysics Details and additional examples can be found in that pap er Elastic waves are waves in an elastic medium such as an iron bar a build ing or the earth and they come in two varieties P waves pressure or primary characterized by longitudinal vibrations and S waves shear or secondary characterized by transverse vibrations The partial dierential equations of elasticity can be written in various forms such as a system of two secondorder equations involving displacements For his numerical simu lations Fornb erg choseaformulation as a system of ve rstorder equations Figures and show the results of calculations for two physical problems In the rst a P wave propagates uninterruptedly through a p erio dic uniform medium In the second an oblique P wave oriented at hits a horizontal interface at whichthewave sp eeds abruptly cut in half The result is reected and transmitted P and S waves For this latter example the actual computation was p erformed on a domain of twice the size shown which is a hint of the trouble one may be willing to go to with sp ectral metho ds to avoid coping explicitly with b oundaries The gures show that sp ectral metho ds may sometimes decisively outp er form secondorder and fourthorder nite dierence metho ds In particular sp ectral metho ds are nondispersive and in a wave calculation that prop erty can be of great imp ortance In these examples the accuracy achieved by the sp ectral calculation on a grid is not matched by fourthorder nite dif ferences on a grid or by secondorder nite dierences on a grid The corresp onding dierences in work and storage are enormous Fornberg picked his examples carefully sp ectral metho ds do not always p erform so convincingly Nevertheless sometimes they are extremely impres sive Although the reasons are not fully understo o d their advantages often hold not just for problems involving smo oth functions but even in the presence of discontinuities The gures in this section will app ear in the published version of this b o ok only with p ermission UNBOUNDED GRIDS TREFETHEN a Schematic initial and end states b Computational results Figure Sp ectral and nite dierence simulations of a P wave propagating through a uniform medium from Fornb erg UNBOUNDED GRIDS TREFETHEN a Schematic initial and end states b Computational results Figure Sp ectral and nite dierence simulations of a P wave incident obliquely up on an interface from Fornb erg UNBOUNDED GRIDS TREFETHEN Unb ounded grids We shall b egin our study of sp ectral metho ds by lo oking at an innite un b ounded domain Of course real computations are not carried out on innite domains but this simplied problem contains many of the essential features of more practical sp ectral metho ds Consider again the set of squareintegrable functions v fv g on the j h unb ounded regular grid hZ As mentioned already in x the foundation of sp ectral metho ds is the sp ectral dierentiation op erator D h h ays One is by means of the which can be describ ed in several equivalent w Fourier transform SPECTRAL DIFFERENTIATION BY THE SEMIDISCRETE FOURIER TRANS Compute v Multiply by i Inverse transform Dv F i F v h h Another is in terms of bandlimited interp olation As describ ed in x one can think of the interp olantasaFourier integral of bandlimited complex exp onentials or equivalentlyasan innite series of sinc functions SPECTRAL DIFFERENTIATION BY SINC FUNCTION INTERPOLATION P Interp olate v by a sum of sinc functions q x v S x x k k h k Dierentiate the interp olant at the grid points x j x Dv q j j Recall that the sinc function S x dened by h sin xh S x h xh UNBOUNDED GRIDS TREFETHEN is the unique function in L that interp olates the discrete delta function e j j e j j and that moreover is bandlimited in the sense that its Fourier transform has compact supp ort contained in h h For higher order sp ectral dierentiation we multiply F v by higher h powers of i or equivalently dierentiate q x more than once tal reason is Why are these two descriptions equivalent The fundamen that S x is not just any interp olant to the delta function e but the band h limited interp olant For a precise argument note that both pro cesses are obviously linear and it is not hard to see that b oth are shiftinvariantin the m m sense that D K v K Dv for any m The shift op erator K was dened can be written as a convolution in Since an arbitrary function v h P sum v v e it follows that it is enough to prove that the two k j k j k pro cesses give the same result when applied to the particular function e That equivalence results from the fact that the Fourier transform of e is the constant function h whose inverse Fourier transform is in turn precisely S x h Since sp ectral dierentiation constitutes a linear op eration on it can h also b e viewed as multiplication by a biinnite To eplitz matrix B C B C B C B C B C B C B C B C B C B C B C D B C h B C B C B C B C B C B C B C B C A As discussed in x this matrix is the limit of banded To eplitz matrices corresp onding to nite dierence dierentiation op erators of increasing orders of accuracy see Table on p In this chapter we drop the subscript on the symbol D used in x We shall be careless in this text ab out the distinction between the op erator D and the matrix D that represents it Another way to express the same thing is to write Dv a v a h UNBOUNDED GRIDS TREFETHEN S x *o h *o S x x h *o o * *o ****oooo ****oooo *o o * *o *o *o a b Figure The sinc function S x and its derivativeS x x h h as in The co ecients of can be derived from either the Fourier transform or the sinc function interpretation Let us b egin with the latter