Some Recent Advances on Spectral Methods for Unbounded Domains


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1 COMMUICATIOS I COMPUTATIOAL PHYSICS Vol. 5, o. 24, pp Commun. Comput. Phy. February 29 REVIEW ARTICLE Some Recent Advance on Spectral Method for Unbounded Domain Jie Shen 1, and LiLian Wang 2 1 Department of Mathematic, Purdue Univerity, Wet Lafayette, I, 4797, USA. 2 Diviion of Mathematical Science, School of Phyical and Mathematical Science, anyang Technological Univerity, , Singapore. Received 13 January 28; Accepted (in revied verion) 8 June 28 Available online 1 Augut 28 Abtract. We preent in thi paper a unified framework for analyzing the pectral method in unbounded domain uing mapped Jacobi, Laguerre and Hermite function. A detailed comparion of the convergence rate of thee pectral method for olution with typical decay behavior i carried out, both theoretically and computationally. A brief review on ome of the recent advance in the pectral method for unbounded domain i alo preented. AMS ubect claification: 6535, 6522,65F5, 35J5 Key word: Spectral method, unbounded domain, orthogonal polynomial, rational function, Hermite function, Laguerre function. Content 1 Introduction Mapped Jacobi method Laguerre pectral method 21 4 Hermite pectral method Implementation, numerical reult and dicuion 23 6 Micellaneou iue and extenion Concluding remark 237 Correponding author. addree: (J. Shen), (L. Wang) 195 c 29 GlobalScience Pre
2 196 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Introduction Spectral method for olving PDE on unbounded domain can be eentially claified into four approache: (i) Domain truncation: truncate unbounded domain to bounded domain and olve the PDE on bounded domain upplemented with artificial or tranparent boundary condition (ee, e.g., [17, 21, 22, 25, 44, 51]); (ii) Approximation by claical orthogonal ytem on unbounded domain, e.g., Laguerre or Hermite polynomial/function (ee, e.g., [7, 14, 2, 3, 31, 36, 43, 47]); (iii) Approximation by other, nonclaical orthogonal ytem (ee, e.g., [14]), or by mapped orthogonal ytem, e.g., image of claical Jacobi polynomial though a uitable mapping (ee, e.g. [32, 34, 35, 54]); (iv) Mapping: map unbounded domain to bounded domain and ue tandard pectral method to olve the mapped PDE in the bounded domain (ee, e.g., [9 12, 15, 24, 26]). Boyd provided in [11] an excellent review on general propertie and practical implementation for many of thee approache. In general, the domain truncation approach i only a viable option for problem with rapidly (exponentially) decaying olution or when accurate nonreflecting or exact boundary condition are available at the truncated boundary. On the other hand, with proper choice of mapping and/or caling parameter, the other three approache can all be effectively applied to a variety of problem with rapid or low decaying (or even growing) olution. Since there i a vat literature on domain truncation, particularly for Helmholtz equation and Maxwell equation for cattering problem and the analyi involved i very different from the other three approache, the domain truncation approach will not be addreed in thi paper. We note that the lat two approache are mathematically equivalent (ee Section for more detail) but their computational implementation are different. More preciely, the lat approach involve olving the mapped PDE (which are often cumberome to deal with) uing claical Jacobi polynomial while the third approach olve the original PDE uing the mapped Jacobi polynomial. The main advantage of the lat approach i that it can be implemented and analyzed uing tandard procedure and approximation reult, but it main diadvantage i that the tranformed equation i uually very complicated which, in many cae, make it implementation and analyi unuually cumberome. On the other hand, we work on the original PDE in the third approach and approximate it olution by uing a new family of orthogonal function which are image of claical Jacobi polynomial under a uitable mapping. The analyi of thi approach will require approximation reult by the new family of orthogonal function. The main advantage i that once thee approximation reult are etablihed, they can be directly applied to a large cla of problem. Thu, we hall mainly concentrate on the econd and third approache, and provide a general framework for the analyi of thee pectral method. While pectral method have been ued for olving PDE on unbounded domain
3 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp for over thirty year, and there have been everal iolated effort in the early year on the error analyi of thee method (ee, e.g. [6, 7, 15, 2, 23, 42]), it i only in the lat ten year or o that the baic approximation propertie of thee orthogonal ytem, and their application to PDE, were ytematically tudied (cf. [13] for a brief account). However, many of thee analye ue different approache and involve complicated Sobolev pace, making it hard for nonexpert to extract ueful information from thee error etimate and to carry out error analyi for their application. The main purpoe of thi paper are three fold: (i) to preent a unified framework, for the analyi of mapped Jacobi, Laguerre and Hermite pectral method, which lead to more concie reult (than thoe appeared in the literature) and optimal approximation reult in mot ituation; (ii) to make a detailed comparion on the convergence rate of different method for everal typical olution; and (iii) to provide a brief (by no mean complete) review on ome of the recent work for the analyi and application of pectral method in unbounded domain. Thi paper i organized a follow. In the next ection, we conider the mapped pectral method and preent a unified framework to tudy their convergence propertie. In Section 3, we conider the approximation by the (generalized) Laguerre polynomial/function, and Section 4 i devoted to the approximation by the Hermite polynomial/function. Thee three ection are preented with a unified tyle and encompa mot of the important approximation reult on thee orthogonal ytem developed in the lat few year. In Section 5, we provide ome implementation detail and compare the performance of different method with two typical example. In Section 6, we dicu variou extenion and other iue related to the application of thee pectral method. We end thi paper with a few concluding remark. We now introduce ome notation. Let ω(x) be a certain weight function in Ω:=(a,b), where a or b could be infinite. We hall ue the weighted Sobolev pace H r ω(ω) (r =,1,2, ), whoe inner product, norm and eminorm are denoted by (, ) r,ω, r,ω and r,ω, repectively. For real r>, we define the pace H r ω(ω) by pace interpolation. In particular, the norm and inner product of L 2 ω(ω) = H ω(ω) are denoted by ω and (, ) ω, repectively. The ubcript ω will be omitted from the notation in cae of ω 1. For notational convenience, we denote k x=d k /dx k, k 1, and for any nonnegative integer, let P be the et of all algebraic polynomial of degree. We denote by c a generic poitive contant independent of any function and, and ue the expreion A B to mean that there exit a generic poitive contant c uch that A cb. 2 Mapped Jacobi method A common and effective trategy in dealing with an unbounded domain i to ue a uitable mapping that tranform an infinite domain to a finite domain. Then, image of claical orthogonal polynomial under the invere mapping will form a et of orthogonal bai function which can be ued to approximate olution of PDE in the infinite domain.
4 198 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Early practitioner of thi approach include Groch & Orzag [24] and Boyd [8]. The book by Boyd [11] contain an extenive review on many practical apect of the mapped pectral method. In the lat couple of year, a erie of paper have been devoted to the convergence analyi of the mapped pectral method (ee, e.g., [34, 35, 39, 54]). We preent below a general framework for the analyi and implementation of the mapped pectral method. To tudy the propertie of the mapped Jacobi approximation, we recall ome baic propertie and reult for the claical Jacobi polynomial J α,β n (y), y I :=( 1,1), n. 2.1 Some reult on Jacobi approximation Let ω α,β (y)=(1 y) α (1+y) β be the Jacobi weight function. For α,β> 1, the Jacobi polynomial are mutually orthogonal in L 2 ω α,β (I), i.e., I J α,β n where δ n,m i the Kronecker function, and (y)jm α,β (y)ω α,β (y)dy=γn α,β δ n,m, (2.1) γn α,β 2 = α+β+1 Γ(n+α+1)Γ(n+β+1) (2n+α+β+1)Γ(n+1)Γ(n+α+β+1). (2.2) They are eigenfunction of the SturmLiouville problem: with the eigenvalue: y ((1 y) α+1 (1+y) β+1 y J α,β n (y))+λn α,β (1 y) α (1+y) β Jn α,β (y)=, (2.3) λ α,β n = n(n+α+β+1), n, α,β> 1. (2.4) ow, we define the L 2 (I) orthogonal proection: ˆπ α,β ω α,β : L2 (I) P ω α,β, uch that Define the weighted pace ( ˆπ α,β v v,v ) ω α,β =, v P. (2.5) ˆB m α,β (I) := {v L2 ω α,β (I) : k y v L2 ω α+k,β+k (I), k m}. (2.6) The following reult wa proved in [19] (ee alo [3, 38]): Lemma 2.1. l y( ˆπ α,β v v) ω α+l,β+l l m m y v ω α+m,β+m, l m, v ˆB α,β m (I). (2.7) Let I α,β be the JacobiGau or JacobiGauRadau interpolation operator. The following interpolation approximation reult can be found in [38]. Lemma 2.2. For any v ˆB α,β m (I) with m 1, y (I α,β v v) α,β ωα+1,β+1+ I v v ω α,β 1 m m y v ω α+m,β+m. (2.8)
5 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Mapping Let u conider a family of mapping of the form: x= g(y;), >, y I :=( 1,1), x Λ :=(,+ )or(,+ ), (2.9) uch that dx dy = g (y;)>, >, y I, g( 1;)=, g(1;)=+, g(±1;)=±, if Λ=(,+ ). if Λ=(,+ ), (2.1) In thi onetoone tranform, the parameter i a poitive caling factor. Without lo of generality, we further aume that the mapping i explicitly invertible, and denote it invere mapping by y= g 1 (x;) := h(x;), x Λ, y I, >. (2.11) Several typical mapping that have been propoed and ued in practice are of the above type (ee, e.g., [11] and the reference therein): Mapping between x Λ=(,+ ) and y I =( 1,1) with >: Algebraic mapping: x= y 1 y 2, y= x x (2.12) Logarithmic mapping: Exponential mapping: x=arctanh(y)= 1+y ln 2 1 y, y=tanh( 1 x). (2.13) x=inh(y), y= 1 ln( x+ x 2 +1 ), y ( 1,1), x ( L,L ), (2.14) where L =inh(). Mapping between x Λ=(,+ ) and y I =( 1,1) with >: Algebraic mapping: x= (1+y) x, y= 1 y x+. (2.15)
6 2 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Logarithmic mapping: ( y+1 ) x=arctanh = 3+y ln y, y=1 2tanh( 1 x). (2.16) Exponential mapping: ( ) x=inh 2 (1+y), y= 2 ln( x+ x 2 +1 ) 1, (2.17) where y ( 1,1) and x (,L ) with L =inh(). The pecial feature which ditinguihe thee mapping i that, a y ±1, x varie algebraically, logarithmically or exponentially for algebraic, logarithmic or exponential mapping, repectively. The parameter i a caling/tretching factor which can be ued to fine tune the pacing of collocation point. We alo notice that the image of the exponential mapping (2.14) and (2.17) i a finite interval, o they combine both mapping and domain truncation. 2.3 Mapped Jacobi approximation Given a mapping x=g(y;) atifying (2.9) (2.11) and a family of orthogonal polynomial {p k (y)} with y I = ( 1,1), { p k (h(x;)) } form a new family of orthogonal function in Λ = (, ) or (, ). For example, the algebraic mapping (2.12) or (2.15) with the Chebyhev or Legendre polynomial lead to orthogonal rational bai function which have been tudied in [8, 9, 14, 34, 35, 4]. For the ake of generality, we conider the mapped Jacobi approximation. Let J α,β k (y) (α, β > 1) be the kth degree claical Jacobi polynomial whoe propertie are ummarized in the Appendix. We define the mapped Jacobi polynomial a,n α,β (x) := Jn α,β (y)= Jn α,β (h(x;)), x Λ, y I. (2.18) We infer from (2.1) that (2.18) define a new family of orthogonal function {,n α,β } in L 2 (Λ), i.e., ω α,β,n α,β (x),m(x)ω α,β α,β (x)dx=γn α,β δ m,n, (2.19) where the contant γ α,β n Λ i given in (2.2), and the weight function ω α,β (x)=ω α,β (y) dy dx = ωα,β (y)(g (y;)) 1 >, (2.2) with y=h(x;) and ω α,β (y)=(1 y) α (1+y) β. We now preent ome approximation propertie of thee mapped Jacobi polynomial. Let u define the finite dimenional approximation pace V α,β, =pan{α,β,n (x) : n=,1,,}, (2.21)
7 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp and conider the orthogonal proection π α,β ( α,β π, u u,v ) Thank to the orthogonality, we can write where, : L2 ω α,β ω α,β ( α,β π, u) (x)= (Λ) V α,β, uch that =, v V α,β,. (2.22) n= û α,β,n = 1 γn α,β u(x) α,β Λ û α,β,n α,β,n (x), (2.23),n (x)ω α,β (x)dx. We now introduce a weighted pace which i particularly uitable to decribe the L 2 proection error. Given a mapping atifying (2.9) (2.11), we et a (x) := dx dy (>), U (y) := u(x)=u(g(y;)). (2.24) The key to expre the error etimate in a concie form i to introduce an operator One verifie readily that du dy = a du dx = D xu, and an induction argument lead to d k U dy k D x u := a du dx. d 2 U dy 2 = a d ( dx a du dx = a d ( d ( ( a } dx dx {{ } k 1 parenthee ) = D 2 xu, du ) ) a := D k dx xu. (2.25) Let u define B α,β m (Λ)={ u : u i meaurable in Λ and u B < α,β m } equipped with the norm and eminorm ( m u B = α,β m k= D k xu 2 ω α+k,β+k ) 1 2, u B m α,β = Dx m u α+m,β+m ω, where the weight function ω α+k,β+k i defined in (2.2). We have the following fundamental reult for the mapped Jacobi approximation.
8 22 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Theorem 2.1. If u B α,β m (Λ), we have that for m, π α,β, u u ω α,β m Dx m u α+m,β+m ω, (2.26) and for m 1, where x (π α,β, u u) ω α,β 1 m Dx m u α+m,β+m ω, (2.27) ω α,β (x)=ω α+1,β+1 (y)g (y;), y=h(x;). Proof. Let U (y) = u(x) = u(h(y;)) whoe Jacobi expanion i U (y) = n=ûα,β n (y). Then, by the definition (2.18), we have the relation between the coefficient of the Jacobi and mapped Jacobi expanion: û α,β,n = 1 γn α,β (u, α,β,n ) α,β ω = 1 γ α,β n,n J α,β (U,Jn α,β ) ω α,β =Û,n α,β. (2.28) Let ˆπ α,β be the L2 orthogonal proection operator aociated with the Jacobi polynomial (cf. (2.5)). By (2.1), (2.19) and Lemma ω α,β 2.1, π α,β, u u 2 ω α,β = (û,n) α,β 2 γ α,β n = (Û α,β n=+1 n=+1,n ) 2 γn α,β = ˆπ α,β U U 2 2m m ω α,β y U 2 ω α+m,β+m 2m Dx m u 2. (2.29) ω α+m,β+m ext, we deduce from (2.18) and the orthogonality of { y Jn α,β } that { x,n α,β } i L orthogonal, and x α,β,n α,β 2 ω = y J α,β n 2 = λ α,β ω α+1,β+1 n γn α,β, where λ α,β n i the eigenvalue of the Jacobi SturmLiouville problem (cf. (2.4)). Therefore, by (2.28) and Lemma 2.1, x (π α,β, Thi end the proof. u u) 2 ω α,β = n=+1 λ α,β n γn α,β (û α,β,n) 2 = n=+1 λ α,β n γn α,β (Û,n α,β ) 2 = y (π α,β U U ) 2 ω α+1,β+1 2(1 m) m y U 2 ω α+m,β+m α,β 2 ω 2(1 m) Dx m u 2. (2.3) ω α+m,β+m
9 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Remark 2.1. It hould be pointed out that under the above general etting, the approximation reult on the higherorder proection, uch a the H 1 (Λ) orthogonal proec ω α,β tion π 1,α,β, : H1 ω α,β (Λ) V α,β,, can be etablihed by uing the exiting Jacobi approximation reult (ee, e.g., [38]) and a imilar argument a above. In particular, applying the above reult with α=β=, 1/2 to the algebraic mapping (2.12) and (2.15) lead to more concie and in ome cae improved, Chebyhev and Legendre rational approximation reult which were developed eparately in [34, 35, 39, 54]. The error etimate in the above theorem look very imilar to the uual pectral error etimate in a finite interval (cf. Lemma 2.1). Firt of all, it i clear from the above theorem that the proection error converge fater than any algebraic rate if a function decay exponentially fat at infinity. For a function with ingularitie inide the domain, the above theorem and Lemma 2.1 lead to the ame order of convergence, auming that the function decay ufficiently fat at infinity. However, for a given mooth function, they may lead to very different convergence rate due to the difference in the norm ued to meaure the regularity. We now determine the convergence rate for three et of function with typical decay propertie: Set 1. Exponential decay with ocillation at infinity u(x)=inkxe x for x (, ) or u(x)=inkxe x2 for x (, ). (2.31) Set 2. Algebraic decay without ocillation at infinity u(x)=(1+x) h for x (, ) or u(x)=(1+x 2 ) h for x (, ). (2.32) Set 3. Algebraic decay with ocillation at infinity u(x)= inkx inkx for x (, ) or u(x)= (1+x) h (1+x 2 for x (, ). (2.33) ) h Conider firt the mapping (2.15). Then, D x = ( dy ) 1 d dx dx = (x+)2 d 2 dx, ( 2 ) k ( 2x ) l 2 ωk,l (x)= x+ x+ (x+) 2. Hence, for u(x)=(1+x) h, it can be eaily checked that Dx mu ω α+m,β+m < if m<2h+α+1, which implie that u π α,β u ω α,β (2h+α+1) (u(x)=(1+x) h ). (2.34)
10 24 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp On the other hand, for u(x) = inkx (1+x) h, it can alo be eaily checked that Dx mu < ω α+m,β+m if m< 2h+α+1 3, which implie that ( u π α,β u ω α,β (2h+α+1)/3 u(x)= inkx ) (1+x) h. (2.35) ext, we conider the mapping (2.12) which lead to ( dy ) 1 d D x = dx dx = (x2 + 2 ) 3/2 2 ω k,l (x)= d dx, ( x x x2 + 2 ) k ( x x x2 + 2 ) l 2 (x ) 3/2. Hence, for u(x)=(1+x 2 ) h, we have Dx m u α+m,β+m ω < if m<2h+α+1, which implie that u π α,β u ω α,β (2h+α+1) (u(x)=(1+x 2 ) h ). (2.36) On the other hand, for u(x)=inkx (1+x 2 ) h, we have Dx m u α+m,β+m ω < if m< 2h+α+1 2, which implie that u π α,β u ω α,β (2h+α+1)/2 ( u(x)= inkx (1+x 2 ) h ). (2.37) A few remark are in order: (i) If h i a poitive integer, then u(x) = (1+x) h and u(x)=(1+x 2 ) h are rational function and they can be expreed exactly by a finite um of mapped rational function; (ii) For other cae, only algebraic convergence rate are achievable even though the function are mooth; (iii) the convergence rate for olution with ocillation at infinitie i much lower than that for olution without ocillation at infinitie; and (iv) For olution with exponential decay at infinity, the convergence rate will be fater than any algebraic rate; numerical reult in [34, 35, 54] (ee alo [11]) indicate that the convergence rate i ubgeometrical a e c ; and (v) numerical reult performed in [34, 35, 39, 54] are conitent with the etimate in (2.34)(2.37). 2.4 Mapped Jacobi interpolation approximation We now conider the Gau and GauRadau quadrature formula on unbounded domain baed on the mapped Jacobi polynomial. To fix the idea, we only conider the Gau quadrature, ince the GauRadau quadrature (which i ueful in the emiinfinite interval) can be treated in exactly the ame fahion. Let { ξ α,β,,},ωα,β be the Jacobi= Gau node and weight, and there hold 1 1 φ(y)ω α,β (y)dy= = φ(ξ α,β, )ωα,β,, φ P 2+1. (2.38)
11 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Applying a mapping (2.9) to the above lead to the mapped JacobiGau quadrature: Λ u(x)ω α,β (x)dx = = u(ζ α,β,, )ρα,β,,, u Vα,β,2+1, (2.39) where ζ α,β,, := g(ξα,β, ;), ρα,β,, := ωα,β,, (2.4) are the mapped JacobiGau node and weight. Accordingly, we can define the dicrete inner product and dicrete norm: (u,v) α,β = ω, u(ζ α,β,, )v(ζα,β,, )ρα,β,,, = u ω α,β =(u,u) 1 2, ω α,β,, u,v C(Λ). The mapped JacobiGau interpolation operator I α,β, : C(Λ) Vα,β,, i defined by I α,β, u Vα,β, α,β uch that (I, u)(ζα,β,, )=u(ζα,β,, ), =,1,,. (2.41) be the JacobiGau (or JacobiGauRadau) interpolation operator. By defini Let I α,β tion, we have I α,β α,β,u(x)=(i U )(y)=(i α,β U )(h(x;)). (2.42) Then, we can eaily derive the following reult by combining Lemma 2.2 and Theorem 2.1. Theorem 2.2. If u B α,β m (Λ) with m 1, then x (I α,β, u u) ω α,β + I α,β, u u ω α,β 1 m Dx m u α+m,β+m ω. (2.43) We now examine how the mapping parameter affect the ditribution of the node. Aume that the node { ζ,,} α,β are arranged in acending order. We firt oberve that = by the mean value theorem, ζ α,β,,+1 ζα,β,, = g (ξ;)(ξ α,β,,+1 ξα,β,, ), (2.44) for certain ξ (ξ α,β,,,ξα,β,,+1 ). Hence, the intenity of tretching eentially depend on the derivative value of the mapping. For the mapping (2.13), (2.12), (2.16) and (2.15), we have dx dy = g (y;)= 1 y 2, (1 y 2 ) 3/2, 2 (3+y)(1 y), repectively. Therefore, the grid i tretched more and more a increae. 2 (1 y) 2, (2.45)
12 26 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp (a) Ditribution of x (, ) n=2 n=16 n=8 n= x (c) Ditribution of x (, ) n=8 n=12 n=16 n=2 (b) Effect of caling factor =2. =.5 =.1 =1. = x (d) Effect of caling factor =2., m=13 =1.5, m=15 =1., m=17 =.5, m=2 =.1, m= x x Figure 1: (a) HermiteGau point ( ) v. mapped LegendreGau point uing the algebraic map (2.12) with =1 ( ) for variou n; (b) Mapped LegendreGau point with n=16 and variou caling factor ; (c) LaguerreGauRadau point ( ) v. mapped LegendreGauRadau point uing the algebraic map (2.15) with =1 ( ) for variou n; (d) Mapped LegendreGauRadau point with n =32 and variou caling factor (m i the number of point in the ubinterval [,1)). In Fig. 1, we plot ample grid ditribution for different caling factor with variou number of node for the mapped Legendre Gau (or GauRadau) point (ee the caption for detail). A comparion with HermiteGau point i alo preented in Fig. 1(a). We notice that the mapped LegendreGau point are more clutered near the origin and pread further, while the HermiteGau point are more evenly ditributed. It hould be oberved that the ditribution of mapped LegendreGau point i more favorable ince a much larger effective interval i covered. However, it can be hown that in both cae,
13 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp the mallet ditance between neighboring point i O( 1 ), a oppoed to O( 2 ) for JacobiGau type node in a finite interval. A comparion of mapped Legendre and Laguerre GauRadau node i hown in Fig. 1(c). The mapped LegendreGauRadau point are much more clutered near the origin, and one can check that the mallet ditance between neighboring point i O( 2 ), a oppoed to O( 1 ) for the Laguerre GauRadau node. Hence, the ditribution of mapped LegendreGauRadau point i more favorable a far a reolution/accuracy i concerned but it will lead to a more retrictive CFL condition if explicit cheme are ued for timedependent problem. 2.5 umerical method uing mapped Jacobi polynomial A generic example Conider the model equation γu x (a(x) x u)= f, x Λ=(,+ ), γ, (2.46) with uitable decay condition at ± which will depend on the weight function in the weighted variational formulation. For a given mapping x=g(y;) with x Λ and y ( 1,1), we recall that the mapped Jacobi polynomial are mutually orthogonal in L 2 ω α,β for (2.46) i to find u V α,β, uch that γ(u,v ) ω α,β + ( a(x) x u, x (v ω α,β ) ) =(I α,β (Λ). Hence, the mapped Jacobi method, f,v ) ω α,β, v V α,β,. (2.47) Let u now conider the econd approach decribed in the introduction. Here, Eq. (2.46) i firt tranformed into γu 1 ( a(g(y;)) ) g (y;) y g (y;) yu = F, (2.48) where U (y)=u(g(y;)) and F (y)= f(g(y;)). Then, let ˆω α,β (y)=ω α,β (y)g (y;), the Jacobi pectral method for (2.48) i to find ũ P uch that ( a(g(y;)) ) γ(ũ,ṽ ) ω α,β+ g (y;) yũ, y (ṽ ˆω α,β ) =(I α,β F,ṽ ) ω α,β, ṽ P. (2.49) One can verify eaily that ũ (y)=u (g(y;)). Hence, the two approache are mathematically equivalent. We remark that the formulation (2.49) i in general more difficult to analyze due to the ingular nature of g (y;), while the analyi for the formulation (2.47) become tandard once we etablih the baic approximation propertie of the mapped Jacobi polynomial.
14 28 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp On the other hand, Eq. (2.48) can be eaily implemented uing the tandard Jacobicollocation (or more pecifically Chebyhevcollocation) method. Indeed, let {h, (y)} 1 be the Lagrange polynomial aociated with the JacobiGau point {y } 1, the Jacobicollocation approximation to (2.48) i to find U, (y) = =1 u h, (y) uch that Let u denote ( 1 ( a(g(y;)) )) γu, (y ) g (y;) y g (y;) yu, (y )= F (y ), 1. (2.5) u=(u 1,,u ) t, f=(f (y 1 ),,F (y )) t, D i = h (y i), D=(D i ), Λ i = a(g(y i;)) g (y i ;), Λ=diag(Λ i), Σ i = 1 g (y i ;), Then, (2.5) lead to the matrix ytem (γi ΣDΛD)u=f, Σ=diag(Σ i). which can be eaily olved by uing a tandard linear algebra routine. ote that in the above procedure, we only need to compute the JacobiGau point {y } 1 and the aociated differentiation matrix D whoe entrie can be found for intance in [19] Error etimate for a model problem We conider the Jacobi rational approximation to the model problem γu(x) 2 xu(x)= f(x), x Λ=(, ), u()=, (2.51) with a uitable decay condition at infinity which i to be determined by the weak formulation of (2.51). For a given mapping, let ω=ω α,β be the weight function aociated with the mapped Jacobi polynomial, and denote We define a bilinear form H 1,ω(Λ)={u H 1 ω(λ) : u()=}. a ω (v,φ)=γ(v,φ) ω +( x v, x (φω)), u,v H 1,ω(Λ). Then, a weak formulation for (2.51) i to find u H,ω 1 (Λ) uch that a ω (u,v)=( f,v) ω, v H 1,ω(Λ), (2.52) for f ( H 1,ω (Λ)). ote that u H 1,ω (Λ) implie a decay condition for u at infinity.
15 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Let u denote X ={u V α,β, :u()=}. We can then define the Galerkin approximation of (2.52) by the mapped Jacobi polynomial a follow: For f L 2 ω(λ) C( Λ), find u X uch that a ω (u,v )=(I α,β, f,v ) ω, v X. (2.53) Unlike the tandard pectral method in a finite domain, the wellpoedne of (2.52) and of (2.53) i not guaranteed for all cae with γ. A general reult for the wellpoedne of an abtract equation of the form (2.52) i etablihed in [49]. For the reader convenience, we recall thi reult below (cf. Lemma 2.3 in [49]): Lemma 2.3. We aume that d 1 =max x Λ ω 1 (x) x ω(x), d 2 =max ω 1 (x) 2 x ω(x) x Λ are finite and that u 2 (x)ω (x) x Λ = for u H 1,ω (Λ). Then, for any u,v H1 ω(λ), we have that a ω (u,v) (d 1 +1) u 1,ω v 1,ω +γ u ω v ω, (2.54) and for any v H 1,ω (Λ), a (ν) ω (v,v) v 2 1,ω +(γ d 2/2) v 2 ω. (2.55) Remark 2.2. The inequality (2.55) i derived under a general framework. For a pecific problem, the contant γ d 2 /2 can often be replaced by a larger contant. Thank to the above lemma, it i then traightforward to prove the following general reult: Theorem 2.3. Aume that the condition of Lemma (2.3) are atified and γ d 2 /2>. Then the problem (2.52) (rep. (2.53)) admit a unique olution. Furthermore, we have the error etimate: u u 1,ω inf v X u v 1,ω + f I α,β, f ω. (2.56) Remark 2.3. With a change of variable x to x/c (c>) for Eq. (2.46), the retriction on γ can be relaxed to γ>. Hence, given a mapping and a pair of Jacobi parameter (α,β), we ut need to compute upper bound for d 1 and d 2, verify that the condition of Theorem 2.3 are atified, and apply the approximation reult in Theorem 2.1 and 2.2 to (2.56) to get the deired error etimate. Conider for example the mapped Legendre method for (2.52) with the mapping (2.15). It can be hown that for thi mapping, we have d 1 2 and d 2 6. Applying Theorem 2.1 and 2.2 to (2.56) with (α,β)=(,) lead to the following reult:
16 21 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Corollary 2.1. Let u and u be the olution of (2.52) and (2.53) with (α,β) = (,) and the mapping (2.15) with =1. Aume that u B, m (Λ), f B, k (Λ) and γ>3. We have u u 1,ω, 1 1 m D m x u ω m,m 1 + k Dx k f ω k,k. (2.57) 1 We note that a lightly improved condition on γ wa derived in [35] uing a refined etimate for (2.55). A imilar procedure can be applied to the mapped Chebyhev method for (2.52) with the mapping (2.15). ote however that in thi cae we have d 1,d 2 =. everthele, one can till how that a ω (, ) i continuou and coercive (cf. [34]). Applying Theorem 2.1 and 2.2 to (2.56) with (α,β)=( 1 2, 1 2 ) lead to the following reult (cf. [34]): Corollary 2.2. Let u and u be the olution of (2.52) and (2.53) with (α,β)=( 1 2, 1 2 ) and the mapping (2.15) with =1. Auming that u B 1/2, 1/2 m (Λ) and f B 1/2, 1/2 k (Λ) and that γ> We have u u 1,ω 1/2, 1/2 1 1 m D m x u ω m 1/2,m 1/2 1 + k Dx k f ω k 1/2,k 1/2. (2.58) 1 Remark 2.4. Error etimate which are eentially equivalent to (2.57) and (2.58) but in different form were derived in [34, 35]. The ame procedure can be ued to derive error etimate on mapped Jacobi method for problem in the whole line (cf. [39, 54]). 3 Laguerre pectral method For problem in a emiinfinite interval, it i natural to ue (generalized) Laguerre polynomial/function which form orthonormal bai in (weighted) Sobolev pace. 3.1 Generalized Laguerre approximation We firt recall ome baic propertie of generalized Laguerre polynomial/function Generalized Laguerre polynomial Let Λ := (, ). The generalized Laguerre polynomial (GLP), denoted by L (α) n (x)(α > 1), are the eigenfunction of the SturmLiouville problem x α e x ( x x α+1 e x x L (α) n (x) ) +λ n L (α) n (x)=, x Λ, (3.1) with the eigenvalue λ n = n. Compared with the Jacobi polynomial in a finite interval, the linear growth of λ n for the Laguerre polynomial indicate, on the one hand, a lower convergence rate of the Laguerre expanion, but on the other hand, lead to better invere inequalitie and conequently milder CFL condition for time dependent problem.
17 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp The GLP are mutually orthogonal in L 2 ω α (Λ) with the weight function ω α (x)=x α e x, i.e., + L (α) n (x)l (α) m (x)ω α (x)dx=γ (α) n δ mn withγ (α) n = Γ(n+α+1). (3.2) Γ(n+1) The threeterm recurrence formula of the GLP read (n+1)l (α) n+1 (x)=(2n+α+1 x)l(α) n (x) (n+α)l (α) n 1 (x), L (α) (x)=1, L (α) 1 (x) = α+1 x. (3.3) We infer from (3.1) and (3.2) that + x L (α) n (x) x L (α) m (x)xω α (x)dx=λ n γ (α) n δ mn. (3.4) An important property of the GLP i the following derivative relation: x L (α) n (x)= L (α+1) n 1 n 1 (x)= L (α) k (x). (3.5) k= The cae α= lead to the claical Laguerre polynomial, which are ued mot frequently in practice and will imply be denoted by L n (x). A in the finite interval cae, it i actually eaier to tudy the whole family of generalized Laguerre polynomial, rather than the Laguerre polynomial alone Approximation reult by generalized Laguerre polynomial We begin by analyzing the approximation propertie of the L 2 ω α orthogonal proection π,α : L 2 ω α (Λ) P, defined by (u π,α u,v ) ωα =, v P. (3.6) It i clear that the polynomial π,α u i the bet approximation u in L 2 ω α (Λ), and with π,α u(x)= n= û (α) n L (α) n (x), û (α) n = 1 + γ (α) u(x)l (α) n (x)ω α (x)dx, n. n Similar to the Jacobi approximation, we define B m α (Λ) := { u : k xu L 2 ω α+k (Λ), k m }, (3.7)
18 212 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp equipped with the norm and eminorm ( m ) 1/2, u B m α = k x u 2 ω u B m α+k α = m x u 2 ω α+m. k= In particular, we omit the ubcript α, when α =. In contrat to the uual weighted Sobolev pace H m ω α (Λ), the weight function correponding to derivative of different order i different in B m α (Λ). We oberve from (3.5) that k xl (α) n (x)=( 1) k L (α+k) n k (x), n k, (3.8) and o { k xl (α) } n are orthogonal with repect to the weight ωα+k, i.e., + k xl (α) l By (3.2) and the Stirling formula, we have Summing (3.9) over k m lead to (x) k xl (α) n (x)ω α+k (x)dx=γ (α+k) n k δ ln. (3.9) Γ(x+1) 2πx x+1/2 e x, x 1, (3.1) γ (α+k) n k = Γ(n+α+1) Γ(n k+1) nα+k, for n 1. m k= ( k x L (α) l, k xl (α) ) n =, ω α+k if l = n and k>min{l,n} which implie that { L (α) } n are orthogonal in the pace B m α (Λ). The fundamental generalized Laguerre approximation reult i tated below (ee, e.g., [19]). Theorem 3.1. For any u B m α (Λ) and m, Proof. Obviouly, we have that l x(π,α u u) ωα+l (l m)/2 m x u ωα+m, l m. (3.11) l x(π,α u u)= û (α) n n=+1 l xl (α) (x). n
19 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Hence, by the orthogonality and the Stirling formula (3.1), l x(π,α u u) 2 ω α+l = Thi complete the proof. n=+1 max n> { γ (α+l) n l û (α) n 2 γ (α+l) n l /γ (α+m) n m l m m x u 2 ω α+m. } γ (α+m) n m û(α) n 2 n=+1 ext, we conider the approximation reult for the H 1 type orthogonal proection. For implicity, we conider only the uual Laguerre cae, i.e., α=. Hereafter, let ω(x)= e x be the uual Laguerre weight function, and denote H 1,ω(Λ)= { u H 1 ω(λ) : u()= }, P ={ φ P : φ()= }. (3.12) Conider the orthogonal proection π 1, : H1,ω (Λ) P, defined by ( (u π 1, u),v ) ω =, v P. (3.13) Theorem 3.2. If u H 1,ω (Λ) and xu B m 1 (Λ), then for m 1, Proof. Let π 1, u u 1,ω 1 2 m 2 m x u ωm 1. (3.14) x φ(x)= π 1, u (y)dy. Then u φ H,ω 1 (Λ). Thank to the imbedding inequality (ee, e.g., [31]) and Theorem 3.1 with α=, we find that Thi end the proof. u ω x u ω, π 1, u u 1,ω φ u 1,ω x (φ u) ω 1 2 m 2 m x u ωm 1. We note that in general Laguerre polynomial are not good candidate for approximation in infinite domain due to their wild behavior at infinity. Thi fact i alo reflected in the error etimate in Theorem 3.1 and 3.2. Although thee error etimate are alo of pectral type, but due to the exponential decay weight in the norm, they only imply meaningful pointwie approximation for a hort interval. Hence, the GLP are only uitable for the approximation of function with fat algebraic (or exponential) growth at infinity. For problem with ome decay propertie at infinity, it i more appropriate to ue the o called generalized Laguerre function (GLF) which we hall conider below.
20 214 J. Shen and L. Wang / Commun. Comput. Phy., 5 (29), pp Generalized Laguerre function The generalized Laguerre function (GLF) are defined by L (α) n (x) := e x/2 L (α) n (x), α> 1, x Λ. (3.15) It i clear that by (3.2), the GLF are orthogonal with repect to the weight function ˆω α = x α, i.e., + L (α) n (x) L (α) m (x) ˆω α (x)dx=γ (α) n δ mn, (3.16) where the contant γ (α) n i given in (3.2). In particular, the uual Laguerre function L n (x)=e x/2 L n (x), n, x Λ, (3.17) are orthonormal with repect to the uniform weight ˆω 1. A in the lat ection, we introduce an operator ˆ x = x which implie that xl (α) n (x)=e x/2 ˆ x L (α) n (x). (3.18) It i traightforward to check that the GLF atify the following propertie: Threeterm recurrence relation (n+1) L (α) n+1 =(2n+α+1 x) L (α) n (n+α) L (α) n 1, L (α) = e x/2, L (α) 1 =(α+1 x)e x/2. The SturmLiouville equation: ( x α e x/2 x x α+1 e x/2 ˆ x L (α) n (x) ) (3.19) +n L (α) n (x)=. (3.2) Orthogonality of the derivative: ˆ x L (α) n (x)ˆ x L (α) m (x) ˆω α+1 (x)dx=λ n γ (α) n δ mn. (3.21) Some recurrence formula: Λ ˆ x L (α) n L (α) n xˆ x L (α) n (x)= L (α+1) n 1 n 1 (x)= L (α) k (x), (3.22a) k= (x)= ˆ x L (α) n (x) ˆ x L (α) n+1 (x), (3.22b) (x)=n L (α) n (x) (n+α) L (α) n 1 (x). (3.22c) We plot in Fig. 2 ome ample graph of GLP and GLF. In contrat to the GLP, the GLF are wellbehaved with the decay property (ee Fig. 2 (d)): L (α) n (x), a x +. (3.23) Therefore, the GLP are uitable for approximation of function which decay at infinity.
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