denote the space of measurable functions v such that is a Hilbert space with the scalar products

Transcription

1 ω Chebyshev Spectral Methods 34 CHEBYSHEV POLYOMIALS REVIEW (I) General properties of ORTHOGOAL POLYOMIALS Sppose I a is a given interval. Let ω : I fnction which is positive and continos on I Let L ω I b be a weight denote the space of measrable fnctions v sch that v ω I v ω d L ω I is a Hilbert space with the scalar prodcts v ω I v d CHEBYSHEV POLYOMIALS are obtained by setting: the weight: ω the interval: I Chebyshev polynomials of degree are epressed as T

2 isin α Chebyshev Spectral Methods 35 CHEBYSHEV POLYOMIALS REVIEW (II) By setting z z, therefore we can derive epressions for the first Chebyshev polynomials we obtain T T T z T z z More generally, sing the de Moivre formla, we obtain z R z z from which, invoing the binomial formla, we get where T m represents the integer part of α m m! m! m! m ote that the above epression is COMPUTATIOALLY USELESS one shold se the formla T instead!

3 Chebyshev Spectral Methods 36 CHEBYSHEV POLYOMIALS REVIEW (III) The trigonometric identity z reslts in the following RECURRECE RELATIO z z z T T T which can be sed to dedce T, based on T and T only Similarly, for the derivatives we get T d dz z dz d d dz z d dz sin z sin z which, pon sing trigonometric identities, yields a RECURRECE RELATIO for derivatives T T T

4 Chebyshev Spectral Methods 37 CHEBYSHEV POLYOMIALS REVIEW (IV) ote that simply changing the integration variable we obtain f ω d π f θ dθ This also provides an isometric (i.e., norm preserving) transformation I ũl, where ũ θ L ω Conseqently, we obtain π θ Tl ω T T T l ωd π θ lθ dθ π c δ l where c if if ote that Chebyshev polynomials are ORTHOGOAL, bt not ORTHOORMAL

5 Chebyshev Spectral Methods 38 CHEBYSHEV POLYOMIALS REVIEW (V) The Chebyshev polynomials T vanish at the GAUSS POITS j defined as j j π j There are eactly distinct zeros in the interval ote that T ; frthermore the Chebyshev polynomials T attain their etremal vales at the the GAUSS LOBATTO POITS j defined as j jπ j There are eactly real etrema in the interval.

6 Chebyshev Spectral Methods 39 CHEBYSHEV POLYOMIALS CLUSTERED GRIDS (I) Interpolation on CLUSTERED GRIDS has very special properties CHEBYSHEV MIIMAL AMPLITUDE THEOREM : Of all polynomials of degree with the leading coefficient (i.e., the coefficient of ) eqal to, the niqe polynomial which has the smallest maimm on is T, the th Chebyshev polynomials divided by. In other words, all polynomials of the same degree and leading coefficient satisfy the ineqality ma P ma T Hence, the TRUCATIO ERROR when given in terms of best behaved T will be Ths, in contrast to interpolation on UIFORM grids, interpolation on CLUSTERED grid is less liely to ehibit the RUGE PHEOMEO ; this concerns clstered grids with asymptotic density of points proportional to π (e.g., varios Chebyshev grids)

7 ω Chebyshev Spectral Methods 4 CHEBYSHEV POLYOMIALS UMERICAL ITEGRATIO FORMULAE (I) FUDAMETAL THEOREM OF GAUSSIA QUADRATURE The abscissas of the point Gassian qadratre formla are precisely the roots of the orthogonal polynomial of order for the same interval and weighting fnction. THE GAUSS CHEBYSHEV FORMULA (eact for ) d π j j with j domain only) j π (the Gass points located in the interior of the Proof via straightforward application of the theorem qoted above.

8 ω ω Chebyshev Spectral Methods 4 CHEBYSHEV POLYOMIALS UMERICAL ITEGRATIO FORMULAE (II) THE GAUSS RADAU CHEBYSHEV FORMULA (eact for ) d π ξ j ξ j jπ with ξ j (the Gass Rada points located in the interior of the domain and on one bondary, sefl e.g., in annlar geometry) Proof via application of the above theorem and sing the roots of the polynomial a Q T a T T a T which vanishes at THE GAUSS LOBATTO CHEBYSHEV FORMULA (eact for ) d π ξ ξ j ξ j jπ with ξ j (the Gass Lobatto points located in the interior of the domain and on both bondaries) Proof via application of the theorem qoted above.

9 Chebyshev Spectral Methods 4 CHEBYSHEV POLYOMIALS UMERICAL ITEGRATIO FORMULAE (III) The GAUSS LOBATTO CHEBYSHEV COLLOCATIO POITS are most commonly sed in Chebyshev spectral methods, becase this set of points also incldes the bondary points (which maes it possible to easily incorporate the BOUDARY CODITIOS in the collocation approach) Using the Gass Lobatto Chebyshev points, the orthogonality relation for the Chebyshev polynomials T and T l with l can be written as Tl ω T T T l ωd π j c j T ξ j Tl ξ j πc δ l where if c if if ote similarity to the corresponding DISCRETE ORTHOGOALITY RELATIO obtained for the trigonometric polynomials

10 Chebyshev Spectral Methods 43 CHEBYSHEV APPROXIMATIO GALERKI APPROACH (I) Consider an approimation of CHEBYSHEV SERIES n I L ω ût Cancel the projections of the residal R fnction (i.e., the Chebyshev polynomials) in terms of a TRUCATED on the first basis Tl ω R T l ω û T T l ω d l Taing into accont the orthogonality condition, epressions for the Chebyshev epansions coefficients are obtained û πc T ωd which can be evalated sing, e.g., the GAUSS LOBATTO CHEBYSHEV QUADRATURES. QUESTIO What happens on the bondary?

11 Chebyshev Spectral Methods 44 CHEBYSHEV APPROXIMATIO GALERKI APPROACH (II) Let P : L ω I be the orthogonal projection on the sbspace polynomials of degree THEOREM For all µ and σ sch that sch that where e µ σ Philosophy of the proof: P µ ω C e µ µ µ σ for µ 3 µ σ for σ of σ, there eists a constant C θ. First establish continity of the mapping ũ, where ũ, from the weighted Sobolev space Hω m I into the corresponding periodic Sobolev space Hp m π π. Then leverage analogos approimation error bonds established for the case of trigonometric basis fnctions σ ω µ θ

12 Chebyshev Spectral Methods 45 CHEBYSHEV APPROXIMATIO COLLOCATIO APPROACH (I) Consider an approimation of L ω I in terms of a trncated Chebyshev series (epansion coefficients as the nnowns) Cancel the residal R on the set of GAUSS LOBATTO CHEBYSHEV collocation points j, j (one cold choose other sets of collocation points as well) ût j û T j j oting that T j j j we obtain jπ jπ and denoting j û π j j The above system of eqations can be written as U TÛ, where U and Û are vectors of grid vales and epansion coefficients, respectively.

13 Chebyshev Spectral Methods 46 CHEBYSHEV APPROXIMATIO COLLOCATIO APPROACH (II) In fact, the matri T is invertible and T j c j c π j j Conseqently, the epansion coefficients can be epressed as follows û c j c j j π j c j c j j R e i π j ote that this epression is nothing else than the COSIE TRASFORMS of U which can be very efficiently evalated sing a COSIE FFT The same epression can be obtained by mltiplying each side of j ût smming the reslting epression from j j by T l j c j to j sing the DISCRETE ORTHOGOALITY RELATIO ξ j ξ j π jct j Tl πc δ l

14 Chebyshev Spectral Methods 47 CHEBYSHEV APPROXIMATIO COLLOCATIO APPROACH (III) ote that the epression for the DISCRETE CHEBYSHEV TRASFORM û c j c j j π j can also be obtained by sing the Gass Lobatto Chebyshev qadratre to approimate the continos epressions û πc T ωd Sch an approimation is EXACT for Analogos epressions for the Discrete Chebyshev Transforms can be derived for other set of collocation points (Gass, Gass Rada) ote similarities with respect to the case periodic fnctions and the Discrete Forier Transform

15 Chebyshev Spectral Methods 48 CHEBYSHEV APPROXIMATIO COLLOCATIO APPROACH (IV) As was the case with Forier spectral methods, there is a very close connection between COLLOCATIO BASED ITERPOLATIO and GALERKI APPROXIMATIO DISCRETE CHEBYSHEV TRASFORM can be associated with an ITERPOLATIO OPERATOR P C : C I defined sch that, j points) PC j j THEOREM Let s constant C sch that for all Hs ω I. (where j are the Gass Lobatto collocation and σ be given and P C σ ω C σ Philosophy of the proof changing the variables to ũ s σ s ω s. There eists a θ convert this problem to a problem already analyzed in the contet of the Forier interpolation for periodic fnctions θ we

16 Tl Tl Chebyshev Spectral Methods 49 CHEBYSHEV APPROXIMATIO COLLOCATIO APPROACH (V) Relation between the GALERKI and COLLOCATIO coefficients, i.e., û e û c πc c j T c j j ω π j d Using the representation lûel Tl in the latter epression and invoing the discrete orthogonality relation we obtain û c ûe c l û e j c j T j j c c û e l l û e C l j c j T j j where C l j c j T j Tl j j c j iπ liπ j c j l iπ l iπ

17 û e Chebyshev Spectral Methods 5 CHEBYSHEV APPROXIMATIO COLLOCATIO APPROACH (VI) Using the identity j piπ p ifm p otherwise m we can calclate C l which allows s to epress the relation between the Galerin and collocation coefficients as follows û c c m m û e m m m û e m The terms in sqare bracets represent the ALIASIG ERRORS. Their origin is precisely the same as in the Forier (psedo) spectral method. Aliasing errors can be removed sing the 3 APPROACH in the same way as in the Forier (psedo) spectral method

18 V X Chebyshev Spectral Methods 5 CHEBYSHEV APPROXIMATIO RECIPROCAL RELATIOS epressing the first Chebyshev polynomials as fnctions of, T T T T T which can be written as V LOWER TRIAGULAR matri X, where T,, and is a Solving this system (trivially!) reslts in the following RECIPROCAL RELATIOS 3 4 T T 3T 4 8 T 3T T T3 4T T4

19 Chebyshev Spectral Methods 5 CHEBYSHEV APPROXIMATIO ECOOMIZATIO OF POWER SERIES Find the best polynomial approimation of order 3 of f Constrct the (Maclarin) epansion e e on Rewrite the epansion in terms of CHEBYSHEV POLYOMIALS sing the reciprocal relations e 8 64 T 9 8 T 3 48 T 4 T 3 9 T 4 Trncate this epansion to the 3 rd order and translate the epansion bac to the representation Trncation error is given by the magnitde of the first trncate term; ote that the CHEBYSHEV EXPASIO COEFFICIETS are mch smaller than the corresponding TAYLOR EXPASIO COEFFICIETS! How is it possible the same nmber of epansion terms, bt higher accracy?

20 Chebyshev Spectral Methods 53 CHEBYSHEV APPROXIMATIO SPECTRAL DIFFERETIATIO (I) Assme the fnction approimation in the form ût First, note that CHEBYSHEV PROJECTIO and DIFFERETIATIO do not commte, i.e., P d d d d Seqentially applying the recrrence relation T T K p c Consider the first derivative P T p û T where, sing the above epression for T coefficients as and û û c p p odd pû p p T where K û T T we obtain, we obtain the epansion

21 Chebyshev Spectral Methods 54 CHEBYSHEV APPROXIMATIO SPECTRAL DIFFERETIATIO (II) Spectral differentiation (with the epansion coefficients as nnowns) can ths be written as ˆÛ where Û û û T, Û Û û û T, and ˆ is an UPPER TRIAGULAR matri with entries dedced based on the previos epression For the second derivative one obtains similarly and û û û c û T p p even p p û p QUESTIO What is the strctre of the second order differentiation matri?

22 Chebyshev Spectral Methods 55 CHEBYSHEV APPROXIMATIO DIFFERETIATIO I REAL SPACE (I) Assme the fnction is approimated in terms of its nodal vales, i.e., j where j are the GAUSS LOBATTO CHEBYSHEV points and C j the associated CARDIAL FUCTIOS where p j C j j Cj j for j for j c j j dt d c j p j m p m T m j for j for j Tm are The DIFFERETIATIO MATRIX p p relating the nodal vales of the p th derivative to the nodal vales of is obtained by differentiating the cardinal fnction appropriate nmber of times p j d p C p d j d p j j

23 Chebyshev Spectral Methods 56 CHEBYSHEV APPROXIMATIO DIFFERETIATIO I REAL SPACE (II) Epressions for the entries of the DIFFERETIATIO MATRIX d GAUSS LOBATTO CHEBYSHEV collocation points j at the the d d d j j j c j c d j j j j 6 Ths in the matri (operator) notation j j j U U ote that ROWS of the differentiation matri are in fact eqivalent to th order asymmetric finite difference formlas on a nonniform grid; in other words, spectral differentiation sing nodal vales as nnowns is eqivalent to finite differences employing ALL GRID POITS AVAILABLE

24 Chebyshev Spectral Methods 57 CHEBYSHEV APPROXIMATIO DIFFERETIATIO I PHYSICAL SPACE (III) (U Epressions for the entries of SECOD ORDER DIFFERETIATIO MATRIX d j U ) at the the GAUSS LOBATTO CHEBYSHEV collocation points d j c j j j j j j j d j j 3 j j 3 j d 3 c 6 d 3 c 6 d d 4 5 ote that d j p d jp d p Interestingly, is not a SYMMETRIC MATRIX...

25 Chebyshev Spectral Methods 58 GALERKI APPROACH BCS VIA BASIS RECOMBIATIO Consider an ELLIPTIC BOUDARY VALUE PROBLEM (BVP) : α α ν a b f β g β g in Chebyshev polynomials do not satisfy homogeneos bondary conditions, hence standard Galerin approach is not directly applicable. BASIS RECOMBIATIO : Convert the BVP to the corresponding form with HOMOGEEOUS BOUDARY CODITIOS (cf. page 7) Tae linear combinations of Chebyshev polynomials to constrct a new basis satisfying HOMOGEEOUS DIRICHLET BOUDARY CODITIOS ϕ ϕ T T T T T even odd ote that the new basis preserves orthogonality

26 Chebyshev Spectral Methods 59 GALERKI APPROACH BCS VIA TAU APPROACH (I) THE TAU METHOD (Lanczos, 938) consists in sing a Galerin approach in which eplicit enforcement of the bondary conditions replaces projections on some of the test fnctions Consider the residal R ν a b f where ût Cancel projections of the residal on the first basis fnctions R Tl ω νû aû bû T T l ωd f T l ωd l Ths, sing orthogonality, we obtain νû aû bû ˆf where ˆf f Tωd

27 β β Chebyshev Spectral Methods 6 GALERKI APPROACH BCS VIA TAU APPROACH (II) oting that T and T, the BOUDARY CODITIOS are enforced by spplementing the residal eqations with α ûg α ûg Epressing û and û in terms of û via the Chebyshev spectral differentiation matrices we obtain the following system ˆ UˆF where Û û T, F ˆf ˆf and the matri is obtained by adding the two rows representing the bondary conditions (see above) to the matri νˆaˆbi. û When the domain bondary is not jst a point (e.g., in D / 3D), formlation of the Ta method becomes somewhat more involved g g

28 b f Chebyshev Spectral Methods 6 COLLOCATIO METHOD (I) Consider the residal R ν a b f where ût Cancel this residal at GAUSS LOBATTO CHEBYSHEV collocation points located in the interior of the domain ν j a j j j j Enforce the two bondary conditions at endpoints α β g α β g ote that this shows the tility of sing the GAUSS LOBATTO CHEBYSHEV collocation points

29 β g Chebyshev Spectral Methods 6 COLLOCATIO METHOD (II) Conseqently, the following system of α α νd j β ad j j d d b g which can be written as cu F, where j c given by with c ν j a eqations is obtained f b c j j j U c j, and the BOUDARY CODITIOS above added as the rows and of c ote that the matri corresponding to this system of eqations may be POORLY CODITIOED, so special care mst be eercised when solving this system for large. Similar approach can be sed when the nodal vales Chebyshev coefficients û are nnowns j, rather than the

30 Chebyshev Spectral Methods 63 CHEBYSHEV METHODS OCOSTAT COEFFICIETS AD OLIEAR EQUATIOS When the eqations has OCOSTAT COEFFICIETS, similar difficlties as in the Forier case are encontered (evalation of COVOLUTIO SUMS ) Conseqently, the COLLOCATIO (psedo spectral) approach is preferable along the gidelines laid ot in the case of the Forier spectral methods Assming aain the elliptic bondary vale problem, we need to mae the following modification to c: where a j d j, j For the Brgers eqation t where each time step W n j Un c t ν j U n ν b U ν we obtain at every time step n U n U n t W n j; ote that an algebraic system has to be solved at

31 Chebyshev Spectral Methods 64 EPILOGUE DOMAI DECOMPOSITIO Motivation: treatment of problem in IRREGULAR DOMAIS STIFF PROBLEMS PHILOSOPHY partition the original domain Ω into a nmber of SUBDOMAIS M Ωm m and solve the problem separately on each those while respecting consistency conditions on the interfaces SPECTRAL ELEMET METHOD consider a collection of problem posed on each sbdomain Ω m L m f m a m m a m m a m m a m Transform each sbdomain Ω m to I se a separate set of m ORTHOGOAL POLYOMIALS to approimate the soltion on every sbinterval bondary conditions on interfaces provide copling between problems on sbdomains