We consider in this paper Legendre and Chebyshev approximations of the linear hyperbolic

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1 LEGEDRE AD CHEBYSHEV DUAL-PETROV-GALERKI METHODS FOR HYPERBOLIC EQUATIOS JIE SHE & LI-LIA WAG 2 Abstract. A Legendre and Chebyshev dual-petrov-galerkin method for hyperbolic equations is introduced and analyzed. The dual-petrov-galerkin method is based on a natural variational formulation for hyperbolic equations. Consequently, it enjoys some advantages which are not available for methods based on other formulations. More precisely, it is shown that (i) the dual-petrov-galerkin method is always stable without any restriction on the coefficients; (ii) it leads to sharper error estimates which are made possible by using the optimal approximation results developed here with respect to some generalized Jacobi polynomials; (iii) one can build an optimal preconditioner for an implicit time discretization of general hyperbolic equations. Dedicated to Professor Ivo Babuska on the occasion of his 80-th birthday. Introduction We consider in this paper Legendre and Chebyshev approximations of the linear hyperbolic equation (.) t u + x (a u) + b u = f, x <, 0 < t T, with given initial data and appropriate non-periodic boundary conditions. There exist a large body of literature on using spectral methods for solving hyperbolic systems (cf. [0, 4, 3, 9] and the references therein). We refer in particular to the recent review paper by Gottlieb and Hesthaven [9] for a up-to-date account on this subject. Previous work can be essentially classified into four different approaches: collocation, Galerkin, tau (cf. [0]) and penalty (cf. [9]). In this paper, we shall take a different point of view by proposing a dual-petrov-galerkin method. The dual-petrov-galerkin method was recently introduced by the first author in [22] for solving third and higher odd-order differential equations. The key idea is to choose the trial functions satisfying the underlying boundary conditions, and the test functions satisfying the dual boundary conditions. This approach enjoys a number of appealing advantages: (i) it leads to a strongly coercive bilinear form despite the fact that the 99 Mathematics Subject Classification. 65M70, 35L50, 65M2. Key words and phrases. Legendre polynomials, Chebyshev polynomials, Spectral and pseudo-spectral approximations, hyperbolic problems, dual-petrov-galerkin method. Department of Mathematics, Purdue University, West Lafayette, I 47907, USA. The work of this author is partially supported by FS grants DMS-0395 and DMS Division of Mathematical Sciences, School of Physical & Mathematical Sciences, anyang Technological University, Singapore

2 2 JIE SHE & LI-LIA WAG leading-order differential operator is not elliptic and non-symmetric. (ii) it leads to a well-conditioned linear system, sparse for problems with constant-coefficients, which can be efficiently solved; (iii) it leads to optimal error estimates. The purpose of this paper is to present a Legendre and Chebyshev dual-petrov- Galerkin method for hyperbolic equations, and to investigate whether their advantages for third and higher-order equations would carry over to first-order equations. The following three issues will be addressed: () Stability: We shall show that the dual-petrov-galerkin method is always stable without any sign restriction on the coefficients a and b. (2) Error analysis: We shall develop new approximation results based on the special basis functions which can be regarded as generalized Jacobi polynomials with index α and/or β. We shall then use these new approximation results to derive sharper error estimates for the dual-petrov-galerkin method. (3) Efficiency: We shall discuss some implementation details of the dual-petrov- Galerkin method in frequency space as well as in physical space. In particular, we shall show that when working in frequency space, the sparse matrix for a problem with constant coefficients can be used as an optimal (independent of number of modes) preconditioner for the full matrix associated with a large class of variable coefficients. The paper is organized as follows. In the next section, we introduce the spectral and pseudo-spectral dual-petrov-galerkin methods in a general setting and prove their stability. In Section 3, we discuss some of the implementation details. Then, in Section 4, we develop sharp approximation results based on special basis functions which are mutually orthogonal in weighted (generalized) Jacobi spaces. We then use these new approximation results to derive, in Section 5, error estimates for the Legendre and Chebyshev spectral dual-petrov-galerkin methods. We conclude with several remarks. We now introduce some notations which will be used throughout the paper. Let χ(x) be a weight function in I = (, ), which is not necessary in L (I). We denote by Hχ(I) r (r = 0,, ) the weighted Sobolev spaces whose inner products, norms and semi-norms are (, ) r,χ, r,χ and r,χ, respectively. In particular, the norm and inner product of L 2 χ(i) = Hχ(I) 0 are denoted by χ and (, ) χ, respectively. The subscript χ will be omitted from the notations in case of χ(x). We denote by ω α,β (x) = ( x) α ( + x) β the Jacobi weight function. In particular, we use ω(x) to denote respectively the Legendre (ω(x) ) or Chebyshev (ω(x) = ( x 2 ) /2 ) weight function. For any non-negative integer, we denote by P the set of all algebraic polynomials of degree. We shall use c to denote a generic positive constant independent of any function and, and we use the expression A B to mean that A cb.

3 HYPERBOLIC EQUATIOS 3 2. The dual-petrov-galerkin method and its stability 2.. An illustrative example. To illustrate the attractive properties of the dual- Petrov-Galerkin method, we first consider the following model equation: (2.) where a is a positive constant. t u + a x u = f, (x, t) I (0, T ], u(, t) = 0, t [0, T ]; u(x, 0) = u 0 (x), x Ī, 2... Variational formulation. Define the dual approximation spaces: (2.2) V = { u P : u( ) = 0 }, V = { v P : v() = 0 }. The Legendre or Chebyshev dual-petrov-galerkin method for (2.) is { Find u (, t) V such that for all t (0, T ], (2.3) ( t u, v )ω + a( x u, v )ω = ( ) f, v ω, v V, with u t=0 = u 0, being a suitable approximation to u 0, and ω(x) being either the Legendre or Chebyshev weight function. x ote that for any v V, we have v +x V. Hence, by setting ω 0(x) = ω(x) x +x, we can rewrite the dual-petrov-galerkin formulation (2.3) in the equivalent (weighted) Galerkin formulation: { Find u (, t) V such that for all t (0, T ], (2.4) ( t u, v )ω 0 + a ( x u, v )ω 0 = ( ) f, v ω 0, v V, with u t=0 = u 0, Stability. The key to stability is the following identities which can be derived directly from an integration by parts: (2.5) (v x, v) ω0 = (v x, v) ω0 = v 2 (x) ( + x) 2 dx, v V (for ω(x) = ), v 2 2 x (x) 2 ( + x) 5 ( x) dx, v V ( for ω(x) = ( x 2 ) /2). Hence, taking v = u in (2.4) leads to that for ω(x) =, 2 t and for ω(x) = x 2, u 2 ω 0(x)dx + a a 2 2 t u 2 ω 0(x)dx + a u 2 2 a 2 u 2 u 2 u 2 ( + x) 2 dx = ( + x) 2 dx + 2a x fu + x dx ( + x) 5 ( x) dx f 2 ( x) 2 dx, x fu ( + x) 3 dx ( + x) 5 ( x) dx + ( x) f 2 3 2a + x dx.

4 4 JIE SHE & LI-LIA WAG The stability of the scheme follows immediately from the above and the Gronwall inequality General setup. We introduce in this subsection a general setup and some notations to be used throughout this paper. Without loss of generality, we conventionally assume that the variable coefficients a and b in (.) satisfy: () a(±, t) does not change sign in [0, T ] ; (2) the functions a, a x, b L (I [0, T ]). To impose boundary conditions, we denote (2.6) Γ := I = {, }, Γ := { x Γ : xa(x, t) < 0 }, Γ + := Γ \ Γ. The problem of interest is of the form ( ) t u(x, t) + x a(x, t)u(x, t) + b(x, t)u(x, t) = f(x, t), (x, t) I (0, T ], (2.7) u(x, t) = g(t), (x, t) Γ [0, T ]; u(x, 0) = u 0 (x), x Ī. More precisely, the boundary conditions are as follows: (i) B u(±, t) = g ± (t), if a(, t) > 0, a(, t) < 0; (2.8) (ii) B u(, t) = g (t), if a(, t) > 0, a(, t) 0; (iii) B u(, t) = g + (t), if a(, t) 0, a(, t) < 0; (iv) B no B.C., if a(, t) 0, a(, t) 0. Since the non-homogeneous boundary conditions can be easily homogenized by subtracting a simple linear function from the exact solution, we shall only consider, without loss of generality, the case g ± (t) = 0. To formulate the dual Petrov-Galerkin formulation uniformly for the four cases, we use the notations {, if Γ {,, if Γ, (2.9) ˆα = ˆβ =, if Γ +,, if Γ +. and define the weight functions (2.0) ω 0 (x) = ω(x)ω ˆα, ˆβ(x), ω (x) = ( x 2 ) ω 0 (x), ω 2 (x) = ( x 2 )ω 0 (x). More precisely, corresponding to each boundary condition (i) B -(iv) B in (2.8), we have (2.) (i) B ˆα = ˆβ = ; (ii) B ˆα =, ˆβ = ; (iii)b ˆα =, ˆβ = ; (iv)b ˆα = ˆβ =. Hereafter, the conditions or expressions labeled by (i) B correspond to the boundary condition (i) B with g ± (t) = 0 in (2.8), and likewise for (ii) B -(iv) B. It will become clear that each time a(±, t) change sign, the variational formulation needs to be changed accordingly.

5 HYPERBOLIC EQUATIOS 5 For each of the boundary conditions (i) B -(iv) B, we define the dual approximation spaces: V = {u P : u(±) = 0}, V = P 2, for (i) B ; (2.2) V = {u P : u( ) = 0}, V = {v P : v() = 0}, for (ii) B ; V = {u P : u() = 0}, V = {v P : v( ) = 0}, for (iii) B ; V = P 2, V = {v P : v(±) = 0}, for (iv) B. One verifies readily that dim(v ) = dim(v ), and v ωˆα, ˆβ V for all v V Spectral approximation. With the above setup, we are ready to formulate the approximation schemes. The Legendre or Chebyshev spectral dual-petrov-galerkin approximation to (2.7) is (2.3) { Find u (, t) V such that for all t (0, T ], ( t u, v )ω + ( x (au ), v )ω + ( bu, v )ω = ( f, v ) ω, v V, with u t=0 = u 0, being a suitable approximation of u 0 (to be specified later). Since for any v V, we have v ω ˆα, ˆβ V, the scheme (2.3) is equivalent to the following weighted Galerkin formulation (notice that ω 0 = ωω ˆα, ˆβ): (2.4) { Find u (, t) V such that for all t (0, T ], ( t u, v )ω 0 + ( x (au ), v )ω 0 + ( bu, v )ω 0 = ( f, v ) ω 0, v V, with u t=0 = u 0,. It will become clear that the dual-petrov-galerkin scheme (2.3) is more suitable for implementation, while the weighted Galerkin formulation (2.4) is more convenient for stability and error analysis. The following coercivity property is essential for the well-posedness of the problems (2.7) and (2.4). Lemma 2.. Let (2.5) A(u, v) = ( x (au), v ) ω 0 + ( bu, v ) ω 0. If u L 2 ω (I) and u x L 2 ω 0 (I), then there exist three real numbers λ i (i = 0,, 2) with λ,λ 2 > 0 such that for t (0, T ], (2.6) λ 0 u 2 ω 0 + λ u 2 ω A(u, u) λ 2 u 2 ω, where the weight functions ω 0 and ω are defined in (2.0).

6 6 JIE SHE & LI-LIA WAG Proof. We first claim that if u L 2 ω (I) and u x L 2 ω 0 (I), then u 2 ω 0 C(Ī). Indeed, for any x, x 2 [, ], we have from the definitions of ω 0 and ω that x2 u 2 (x 2 )ω 0 (x 2 ) u 2 (x )ω 0 (x ) = x (u 2 ω 0 )dx x x2 x2 2 x u u ω 0 dx + u 2 x ω 0 dx x x2 x x ( x u 2 + u 2 )ω 0 dx + x2 x x u 2 ω 0 dx + x2 x x2 x u 2 ω dx. u 2 ω dx Here, we used the inequalities: ω 0 ω and x ω 0 ω. Hence, letting x 2 x, we find that u 2 ω 0 C(Ī). Thanks to this fact and u L2 ω (I), i.e., u 2 (x)ω 0 (x) x 2 dx < +, we have that u 2 (x)ω 0 (x) 0 as x. Integration by parts yields that (2.7) A(u, u) = where ( x a + b)u 2 ω 0 dx + 2 a x (u 2 )ω 0 dx = (2.8) s(x, t) = 2 xa + b 2 aω 0 xω 0. More precisely, we have from (2.0) that in the Legendre case (ω = ): 2 xa + b xa, for (i); x2 2 xa + b + a, for (ii); x2 (2.9) s(x, t) = 2 xa + b a, for (iii); x2 2 xa + b + xa, for (iv), x2 and in the Chebyshev case (ω = x 2 ): (2.20) s(x, t) = 2 xa + b 3 xa, 2 x2 for (i); 2 xa + b + (2 x)a, 2 x2 for (ii); 2 xa + b (2 + x)a, 2 x2 for (iii); 2 xa + b + xa, 2 x2 for (iv). su 2 ω 0 dx

7 HYPERBOLIC EQUATIOS 7 ext, by an argument similar to the proof of Lemma. in [4], we prove that there exist three constants λ i (i = 0,, 2) with λ, λ 2 > 0 such that (2.2) λ 0 + λ x 2 s(x, t) λ 2, (x, t) I [0, T ]. x2 Indeed, the inequality at the right-hand side in the above relation is obvious so we only need to prove the one at the left-hand side. Let us consider for instance the case (i) in (2.9). Indeed, by the continuity of a(x, t) and the condition that a(±, t) is bounded away from zero on [0, T ], we know that there exists δ (0, ) such that xa x 2 λ x 2 with λ > 0 and x I \ [ δ, δ]. Then, since x a and b are bounded in I, and xa x 2 is bounded in [ δ, δ], we infer (2.2). The other cases can be proved in a similar way. Finally, the desired result follows from (2.7) and (2.2). Remark 2.. We may even require that the constant λ 0 0. Indeed, if λ 0 < 0, the change of variable u e λ 0t u in (2.7) leads to the same equation with b(x) replaced by b(x) λ 0 (cf. [4]). Hence, the transformed equation will satisfy s(x, t) λ x 2, x I. We note that the existence and uniqueness of solutions for (2.7) in the weighted Sobolev spaces that we consider here have been established recently in [3]. We provide below an a priori estimate which is an immediate consequence of Lemma 2.: Theorem 2.. Let u and u be respectively the solutions of (2.7) and (2.4). If u 0 L 2 ω 0 (I) and f L 2 (0, T ; L 2 ω 2 (I)), then we have (2.22) u L (0,T ;L 2 ω 0 (I)) + λ u L 2 (0,T ;L 2 ω (I)) u 0 ω0 + f L 2 (0,T ;L 2 ω 2 (I)), and (2.23) u L (0,T ;L 2 ω 0 (I)) + λ u L 2 (0,T ;L 2 ω (I)) u 0, ω0 + f L 2 (0,T ;L 2 ω 2 (I)), where the weights ω i (i = 0,, 2) are defined in (2.0). Proof. Taking the inner product of the first equation of (2.7) with ω 0 u, and using the fact: ω 2 0 = ω ω 2, we derive from the Cauchy-Schwarz inequality that 2 t u 2 ω 0 + A(u, u) = (f, u) ω0 λ 2 u 2 ω + 2λ f 2 ω 2. Hence, by Lemma 2., (2.24) 2 t u 2 ω 0 + λ 2 u 2 ω λ 0 u 2 ω 0 + 2λ f 2 ω 2. Consequently, using the Gronwall inequality leads to (2.22). Following exactly the same procedure, one can prove (2.23). Remark 2.2. The definition of ˆα and ˆβ is different from e and e + defined in (.3) of [4]. Hence, although the approach in this section is similar to that in [4], our variational formulation, and therefore our scheme as well as its stability and convergence properties,

8 8 JIE SHE & LI-LIA WAG is different from theirs. In particular, Theorem 2. holds without assuming that the coefficients a and b satisfy any coercivity condition such as (.2) in [4]. We note that by assuming a(x) > 0, a different set of stability conditions involving the weight and a(x) is derived recently in [6] Pseudo-spectral approximation. In practical implementations, the continuous inner product (, ) ω should be replaced by a discrete inner product,,ω (pseudospectral) which is based on a suitable Gaussian-type quadrature. Let,,ω be the discrete inner product associated, respectively, with the Gauss- Lobatto, Gauss-Radau (with x 0 = ), Gauss-Radau (with x = ) and Gauss quadrature for the four different boundary conditions (i) B, (ii) B, (iii) B and (iv) B. Let I : C(I) P be the corresponding interpolation operator. We recall that (cf. [3]) (2.25) u, v,ω = (u, v) ω, u v P 2+δ, where δ =, 0, for Gauss, Gauss-Radau and Gauss-Lobatto, respectively. It is well-known that one needs to use the skew-symmetric form in a pseudo-spectral approximation to ensure the theoretical stability for general situations (see [7, 4, 9]). We denote (2.26) A (u, v ) = a x u + x I (au ), v 2,ω + ( 2 xa + b)u, v,ω. The skew-symmetric Legendre or Chebyshev pseudo-spectral dual-petrov-galerkin approximation to (2.7) is: { Find u (, t) V such that for all 0 < t T, (2.27) t u, v,ω + A (u, v ) = f, v,ω, v V. Lemma 2.2. Let λ i (i = 0,, 2) be the same as in Lemma 2.. Then, (2.28) λ 0 u 2 ω 0 + λ u 2 ω A (u, u ω ˆα, ˆβ) λ 2 u 2 ω, u V, where ω ˆα, ˆβ, ω 0 and ω correspond to any of the boundary conditions in (2.8). Proof. The proof is essentially the same as that of Lemma 2.. Thanks to (2.25), we only need to show (2.29) (λ0 + λ x 2 )u, u ω ˆα, ˆβ,ω A (u, u ω ˆα, ˆβ) λ 2 x 2 u, u ω ˆα, ˆβ,ω. Indeed, we derive from (2.25) and integration by parts that a x u, u ω ˆα, ˆβ,ω = x u, I (au )ω ˆα, ˆβ,ω = x u, I (au )ω ˆα, ˆβ ω = ( ) u, x I (au )ω 0 + I (au ) x ω 0 which along with (2.26) implies that = x I (au ), u ω ˆα, ˆβ,ω aω 0 xω 0 u, u ω ˆα, ˆβ,ω, (2.30) A (u, u ω ˆα, ˆβ) = ( 2 xa + b 2 aω 0 xω 0 )u, v,ω.

9 HYPERBOLIC EQUATIOS 9 Hence, (2.29) is a directly consequence of (2.9). Thanks to the above lemma, we have immediately the following stability result: Theorem 2.2. If u 0, L 2 ω 0 (I) and f L 2 (0, T ; L 2 ω 2 (I)), then we have (2.3) u L (0,T ;L 2 ω 0 (I)) + λ u L 2 (0,T ;L 2 ω (I)) u 0, ω0 + I f L 2 (0,T ;L 2 ω 2 (I)). 3. Implementations In this section, we discuss some of the implementation details of the dual-spectral- Galerkin method. We note that the slightly more costly skew-symmetric form may not be necessary since the standard form is often numerically stable, at least for well-resolved problems (cf. [, 8, 9]). Hence, we present two implementations below, one uses the skew-symmetric form (2.27) with basis functions in frequency space, and the other is the standard pseudo-spectral form with basis functions in physical spaces. 3.. Implementations in frequency space. To simplify the presentation, we shall only provide details for the second boundary condition in (2.8). The other cases can be treated in a similar fashion. As demonstrated in [20, 2], it is advantageous to use basis functions which are compact combinations of the Legendre and Chebyshev polynomials. Therefore, we set in the Legendre case φ k (x) = L k (x) + L k+ (x), ψ k (x) = L k (x) L k+ (x), and in the Chebyshev case φ k (x) = ( + x)t k (x), ψ k (x) = ( x)t k (x), where L k (x) and T k (x) are respectively the Legendre and Chebyshev polynomials of degree k. Then, we have for the second case in (2.2), (3.) V = span { φ 0, φ,, φ }, V = span { ψ 0, ψ,, ψ }. Therefore, setting (3.2) f j = f, ψ j,ω, f = (f0, f,, f ) t, u = k=0 ũ k φ k, ũ = (ũ 0, ũ,, ũ ) t, m jk = φ k, ψ j,ω, M = (m jk ) j,k=0,,,, s jk = 2 aφ k + xi (aφ k ), ψ j,ω, S = (s jk ) j,k=0,,,, q jk = ( 2 xa + b)φ k, ψ j,ω, Q = (q jk ) j,k=0,,,, we find that (2.27) becomes the following system of ODEs: (3.3) Mũ t + (S + Q)ũ = f.

10 0 JIE SHE & LI-LIA WAG We recall that the discrete inner product,,ω here is associated with the Gauss-Radau interpolation nodes with x 0 =. To avoid severe restrictions on time step associated with explicit time discretizations of spectral methods, we shall consider implicit schemes for integrating (3.3). After discretizing (3.3) by a suitable implicit scheme, we need to solve at each time step the following linear system: (3.4) ( αm + S + Q )ũ = g, where α = O( t ) is a constant. It follows from the orthogonality of Legendre and Chebyshev polynomials that m jk = 0 for j k >. Hence, the mass matrix M is (non-symmetric) tridiagonal and its entries can be easily determined. On the other hand, the matrices S and Q are usually full, unless a and b are simple, low-order polynomials in x Case : a(x, t) ā and b(x, t) b are two constants. In this simple case, we have from (4.6) and (A.3) that for ω(x), (3.5) while for ω(x) = x 2, (3.6) Let us denote (3.7) Then, we can rewrite (3.4) as s jk = ā φ k, ψ j,ω = ā (φ k, ψ j) ω = 2āδ jk, q jk = b φ k, ψ j,ω = 0, j k >, s jk = ā φ k, ψ j,ω = ā (φ k, ψ j) ω = 0, j k >, q jk = b φ k, ψ j,ω = 0, j k > 2. s 0 jk = φ k, ψ j,ω, q 0 jk = φ k, ψ j,ω, S 0 = (s 0 jk ) j,k=0,,,, Q 0 = (q 0 jk ) j,k=0,,,. (3.8) (αm + ās 0 + bq 0 )ũ = g, which can be efficiently inverted. As we demonstrate below, the linear system in this simple case can be used as an effective preconditioner for the general case Case 2: variable coefficients. As observed above, for general variable coefficients a and b, the matrices S and Q are full. Hence, a direct inversion of (3.4) is not advisable. However, we shall use Lemma 2.2 to show that (3.4) can be solved effectively by using a preconditioned iterative method. Since φ k ω, V, there exists a unique set of {h kj} such that (3.9) φ k ω, = j=0 h kj ψ j, k = 0,,,.

11 HYPERBOLIC EQUATIOS We denote H = (h kj ) k,j=0,, and for v = (v 0, v,, v ) t, we define v, v l 2 := j=0 v2 j which is the inner product in l2. Let u, {ũ j } and ũ be the same as before, we have (3.0) HS 0 ũ, ũ l 2 = = k,j,l=0 l=0 ũ k h kj s 0 jlũl = ũ l φ l, k,j=0 k,j,l=0 ũ k h kj φ l, ψ j,ω ũ l ũ k h kj ψ j =,ω l=0 = x u, ω, u,ω = ( xu, u ) ω0. ũ l φ l, k=0 ũ k φ k ω,,ω Therefore, by (2.5), HS0 ũ, ũ = u l 2 2 ω, for ω ; (3.) 2 u 2 ω HS 0 ũ, ũ 3 l 2 2 u 2 ω, for ω =. x 2 Similarly, (3.2) H(αM + S + Q)ũ, ũ = = j=0 j=0 l 2 = k,j,l=0 ũ k h kj (αm jl + s jl + q jl )ũ l ũ j [ 2 (aφ j + x I (aφ j )) + (α + 2 xa + b)φ j ], k,j=0 ũ j [ 2 (aφ j + x I (aφ j )) + (α + 2 xa + b)φ j ], k=0 = 2 (a xu + x I (au ) + (α + 2 xa + b)u, u ω,,ω. Hence, thanks to Lemma 2.2, ũ k h kj ψ j,ω ũ k φ k ω,,ω (α + λ 0 ) u 2 ω 0 + λ u 2 ω H(αM + S + Q)ũ, ũ l 2 α u 2 ω 0 + λ 2 u 2 ω. Assuming α + λ 0 0 (which holds for most cases since α = O( t ), see Remark 2. otherwise), we derive from the above, (3.) and the fact ω 0 ω that (3.3) 2 λ HS0 ũ, ũ H(αM + S + Q)ũ, ũ 3 l 2 l 2 2 (α + λ 2) HS 0 ũ, ũ. l 2 The relation (3.3) indicates that a good preconditioner for αm + S + Q is S 0. In fact, a more robust preconditioner is (βm + S 0 ) for some β 0 with β α for α. In Table, we list the number of BCG iterations needed to achieve six-digit accuracy for solving (3.4) in the Legendre case for the following two test examples using (βm + S 0 ) as preconditioner: Example I. a(x) = 2 + sin(2πx), b(x) = 2π cos(2πx). (a(x) does not change sign in (, ))

12 2 JIE SHE & LI-LIA WAG Table. number of BCG iterations needed with (βm + S 0 ) as preconditioner Example I with α =, β = 0 Example I with α = β = 00 Example II with α =, β = 0 Example II with α = β = 00 Example II. a(x) = + 2 sin(2πx), b(x) = 4π cos(2πx). (a(x) changes sign in (, )) ote that in the Legendre case, we have S 0 = 2I. Hence, no preconditioner is needed in this case if we take β = 0. We observe from Table that (i) for Example I where a(x) does not change sign, the preconditioner is very robust for both α = and α = 00, and (ii) for Example II where a(x) changes sign, the preconditioner is only robust when α is sufficiently large and β α. This is consistent with (3.3) where it is assumed that α + λ 0. We note however that although the preconditioner built in the frequency space is quite robust with respect to, it may not be robust with large variations of a and b. For the latter case, it may be advantageous to use an implementation in physical space which we shall discuss below Implementations in physical space. Let {x j } j=0 (with x 0 = ) be the set of Legendre or Chebyshev Gauss-Radau points. We set x + =. Hence, replacing (, ) ω by the discrete inner product,,ω in (2.3), the standard pseudo-spectral dual- Petrov-Galerkin method is: { Find u (, t) V such that for all t (0, T ], (3.4) t u, v,ω + x (au ) + bu, v,ω = f, v,ω, v V. Let ˆφ j (x) P be the Lagrange polynomial associated with {x j } j=0 such that ˆφ j (x k ) = δ kj for j, k = 0,,,, and let ˆψ j (x) P be the Lagrange polynomial associated with {x j } + j= such that ˆψ j (x k ) = δ kj for j, k =, 2, +. Then, (3.5) V = span { ˆφ, ˆφ2,, ˆφ }, V = span { ˆψ, ˆψ2,, ˆψ },

13 HYPERBOLIC EQUATIOS 3 and we have w jk := ˆφ k, ˆψ j,ω = δ jk ρ j, (3.6) s jk := x (a ˆφ k ), ˆψ j,ω = ( a(x j, t) ˆφ k (x j) + x a(x k, t)δ kj ) ρj + a(, t) ˆφ k ( ) ˆψ j ( )ρ 0, q jk := b ˆφ k, ˆφ j,ω = b(x k, t)δ kj ρ j, f, ψ j,ω = f(x j, t)ρ j + f(, t) ˆψ j ( )ρ 0. Let us denote W = diag(ρ, ρ 2,, ρ ), S = (s jk ) j,k=,2,,, Q = (q jk ) j,k=,2,,, u = u (x j, t) ˆφ k (x), u = ( u(x, t), u(x 2, t),, u(x, t) ) t, (3.7) k= Then, (3.4) becomes g = ( ˆψ ( ), ˆψ 2 ( ),, ˆψ ( ) ) t, f = ( f(x, t), f(x 2, t),, f(x, t) ) t. (3.8) W u t + (S + Q)u = W f + f(, t)ρ 0 g. In a pointwise form, the above equation, after inverting the diagonal matrix W, can be written as: Find u P for all t (0, T ] such that t u (x j, t) + x (a(x, t)u (x, t)) x=xj + b(x j, t)u (x j, t) (3.9) = f(x j, t) + ψ j ( ) ρ 0 ( f(, t) a(, t)u ρ (, t)), j, j u (, t) = 0. Hence, the dual-petrov-galerkin method does not correspond to a pure collocation method, instead, it is a pseudo collocation scheme with an additional boundary residual term on the right-hand side. In Figure, we plot the eigenvalue distribution of W S for various, where W and S are the matrices defined in (3.7) with a(x) =. It is clear that for all, the real parts of the eigenvalues are always positive, indicating the good stability of our dual-petrov formulation. Remark 3.. One may solve (3.8) using a suitable explicit scheme, which will be subjected to a usual CFL constraint (see, for instance, [0, 2]). On the other hand, since (3.8) was derived from a proper variational formulation with a coercive bilinear form, it may be possible, as in the case of elliptic equations (cf. [8, 5, 9]), to build an optimal finite element preconditioner which is robust with respect to both the number of points and the large variations of coefficients a and b. However, this subject and a detailed study on the robustness of the preconditioning in frequency space is beyond the scope of this paper and will be investigated elsewhere.

14 4 JIE SHE & LI-LIA WAG 4. Error Estimates In this section, we shall present some optimal Legendre and Chebyshev approximation results measured in strong norms, and perform the error analysis for the proposed dual- Petrov-Galerkin schemes. 4.. Legendre case (ω = ). We first introduce the basis functions for the dual spaces in (2.2). Let (4.) ϕ n (x) = L n+2 (x) L n (x), φ n (x) = L n (x) + L n+ (x), ψ n (x) = L n (x) L n+ (x). Thanks to the fact: L k (±) = (±) k, one verifies that ϕ n (±) = φ n ( ) = ψ n () = 0. Let J α,β k (α, β > ) be the classical Jacobi polynomial of degree k (see Appendix A for its properties). The following identities hold (see Appendix B for the proof): (4.2) ϕ n (x) = 2n + 3 2(n + ) ( x2 )J, n (x), x ϕ n (x) = (2n + 3)L n+ (x); (4.3) φ n (x) = ( + x)jn 0, (x), x φ n (x) = (n + )Jn,0 (x); (4.4) ψ n (x) = ( x)jn,0 (x), x ψ n (x) = (n + )Jn 0, (x). The orthogonality of Jacobi polynomials (cf. (A.)) implies that {ϕ n }, {φ n } and {ψ n } are mutually orthogonal in L 2 ω, (I), L 2 ω 0, (I) and L 2 ω,0 (I), respectively. Hence, {ϕ n }, {φ n } and {ψ n } can be viewed as extensions of the classical Jacobi polynomials to the cases with parameters (α, β) = (, ), (α, β) = (0, ) and (α, β) = (, 0), respectively. One also observes that (4.5) x ϕ n (x)l m+ (x)dx = (4.6) I I I ϕ n (x) x L m+ (x)dx = 2δ n,m, x φ n (x)ψ m (x)dx = φ n (x) x ψ m (x)dx = 2δ n,m. I Besides, as shown in Appendix B, we have (4.7) xϕ l m (x) xϕ l n (x)ω l,l (x)dx = µ () n,l δ m,n, n l 0; (4.8) (4.9) with I I I l xφ m (x) l xφ n (x)ω l,l (x)dx = µ (2) n,l δ m,n, n l 0; l xψ m (x) l xψ n (x)ω l,l (x)dx = µ (2) n,l δ m,n, n l 0, (4.0) µ () + l + ) n,l = 2(2n + 3)Γ(n Γ(n l + 3), + l + ) µ(2) n,l = 2Γ(n Γ(n l + 2).

15 HYPERBOLIC EQUATIOS Legendre approximations. We first notice that the polynomial space V in (2.2) for cases (i) B -(iv) B are respectively identical to (i) B V, = span { ϕ k : 0 k 2 }, (ii) B V 0, = span { φ k : 0 k }, (iii) B V,0 = span { ψ k : 0 k }, (iv) B V 0,0 = span{ L k : 0 k 2 }. For each pair (α, β) listed below, (4.) (i) B α = β =, (ii) B α = 0, β =, (iii) B α =, β = 0, (iv) B α = β = 0, we define the weighted Sobolev space (4.2) B r α,β (I) = { u : l xu L 2 ω α+l,β+l (I), 0 l r }, r, equipped with the norm and semi-norm ( r ) u B r α,β = xu l 2 2 ω, u α+l,β+l B r α,β = xu r ω α+r,β+r. l=0 Consider the orthogonal projection π α,β : L2 (I) V α,β ω α,β ( (4.3) π α,β u u, v Let us first present a very special property of π α,β. defined by )ω α,β = 0, v V α,β. Lemma 4.. For each pair (α, β) in (4.), let (ˆα, ˆβ) be the corresponding pair defined in (2.). Then (4.4) and (4.5) ( x (π α,β u u), v ωˆα, ˆβ ) = 0, v V α,β, ( x (π α,β u u), v ) = 0, v V. Proof. For each case of (i) B -(iv) B, we notice that ( π α,β u u) ˆα, ˆβ v ω = 0, ( ωα,β x v ω ˆα, ˆβ ) V α,β. Hence, (4.4) is a direct consequence of an integration by parts and the definition (4.3). Since any v V can be expressed as v = vωˆα, ˆβ with v V α,β, (4.5) follows from (4.4). We are now in position to state the main approximation properties of these orthogonal projection operators. Theorem 4.. For each pair of (α, β) in (4.), and for any u Bα,β r (I) with r, (4.6) x m ( π α,β u u) ω α+m,β+m m r xu r ω α+r,β+r, 0 m r. Proof. The result with α = β = 0 is well-known (see, for instance, [7, 2]). A more general result (for all α and β which are integers) was presented in [5] but the proof was omitted due to space limitation. For the readers convenience, we provide below a detailed proof for (4.6) with (i) α = β =. The other cases can be proved similarly.

16 6 JIE SHE & LI-LIA WAG For any u L 2 ω (I), we write, (4.7) u(x) = û n ϕ n (x), with û n = n=0 µ () n,0 So formally we have from (B.) and (A.) that (4.8) xu l 2 ω = κ 2 l,l n,lû2 nγ l,l n l+2 = On the other hand, Hence, by (4.2) and (4.8), π, n=l u(x) u(x) = m x (π, u u) 2 ω l,l = C m,r n= n= n= where by (4.0) and the Stirling formula (see [6]), µ () n,m C m,r = max n µ () n,r (u, ϕ n ) ω,. n=l û n ϕ n (x). û 2 nµ () n,m µ () n,lû2 n. µ () n,rû 2 n C m,r r xu 2 ω r,r 2(m r) Convergence of (2.3) with ω. Let u and u be the solutions of (2.7) and (2.3) (with ω ), respectively, and set ê = π α,β u u, e = u u = (u π α,β u) + ê. Theorem 4.2. Let u t=0 = u 0, = π α,β u 0. For each of the pair (α, β) in (4.), assuming u L 2 (0, T ; L 2 ω (I)) L (0, T ; Bα,β r (I)) and tu L 2 (0, T ; B r α,β (I)) with integer r, we have (4.9) u u L (0,T ;L 2 ω 0 (I)) r( t x r u L 2 (0,T ;L 2 ωα+r,β+r (I)) + xu r ) L (0,T ;L 2. ω α+r,β+r (I)) where ω 0 and ω are given in (2.0). Proof. By (2.7) and (2.4), ( t ê, v ) ω0 + A(ê, v ) = ( t (π α,β u u), v ) (4.20) + ( x (a(π α,β u u)), v )ω 0 + ( b(π α,β u u), v ω 0 ) ω 0, v V. Taking v = ê in (4.20) and using (2.23) (note: ê (0) = 0), we find that (4.2) ê L (0,T ;L 2 ω 0 (I)) + λ ê L 2 (0,T ;L 2 ω (I)) G + G 2

17 HYPERBOLIC EQUATIOS 7 with G = t (π α,β u u) L 2 (0,T ;L 2 ω 2 (I)), G 2 = x (a(π α,β u u)) + b(πα,βu u) L 2 (0,T ;L 2 ω 2 (I)). By (4.6) with m = 0,, and the definition of ω 2 ( ω α,β ) in (2.0), G t (π α,β u u) L 2 (0,T ;L 2 ω α,β (I)) r t x r u L 2 (0,T ;L 2 ωα+r,β+r (I)), and using the fact a, a x, b L (I (0, T ]), G 2 π α,β u u L 2 (0,T ;Bα,β (I)) r xu r L 2 (0,T ;L 2 ωα+r,β+r (I)). Hence, plugging the estimates of G and G 2 into (4.2) leads to (4.22) ê L (0,T ;L 2 ω 0 (I)) + λ ê L 2 (0,T ;L 2 ω (I)) r ( t r x u L 2 (0,T ;L 2 ω α+r,β+r (I)) + r xu L 2 (0,T ;L 2 ω α+r,β+r (I)) ). Since in the Legendre case, ω 0 = ω ˆα, ˆβ ω α,β (cf. (2.) and (4.)), using (4.6) (with m = 0) and (4.22) yields that u u L (0,T ;L 2 ω 0 (I)) π α,β u u L (0,T ;L 2 ω 0 (I)) + ê L (0,T ;L 2 ω 0 (I)) π α,β u u L (0,T ;L 2 ω α,β (I)) + ê L (0,T ;L 2 ω 0 (I)) r ( t r x u L 2 (0,T ;L 2 ω α+r,β+r (I)) + r xu L (0,T ;L 2 ω α+r,β+r (I)) ). This completes the proof. Remark 4.. If a(x) is a constant, the above result can be slightly improved. Indeed, when taking v = ê in (4.20) in this case, thanks to Lemma 4., we find that the term corresponding to G 2 becomes G 2 = b(π α,β u u) L 2 (0,T ;L 2 ω 2 (I)). Therefore, and G 2 max x Ī b(x) π α,β u u L 2 (0,T ;L 2 ω 2 (I)) r xu r L 2 (0,T ;L 2 ωα+r,β+r (I)), Hence, we have G r t xu r L 2 (0,T ;L 2 ωα+r,β+r (I)). u u L (0,T ;L 2 ω 0 (I)) r( r x t u L 2 (0,T ;L 2 ω α+r,β+r (I)) + r xu L 2 (0,T ;L 2 ω α+r,β+r (I)) ) Chebyshev case (ω = ( x 2 ) /2 ). In this subsection, we perform the error analysis of the Chebyshev dual-petrov-galerkin scheme (2.3) with V /V (i) B -(iv) B in (2.2). given by

18 8 JIE SHE & LI-LIA WAG Chebyshev approximations. First, we establish an embedding result associated with the weights given by (2.0) (2.): (4.23) ω 0 (x) = ω(x)ω ˆα, ˆβ(x) = ω 3/2, 3/2, ω /2, 3/2, ω 3/2,/2, ω /2,/2, ω (x) = ( x 2 ) ω 0 (x) = ω 5/2, 5/2, ω /2, 5/2, ω 5/2, /2, ω /2, /2, for the cases (i) B -(iv) B, respectively. Lemma 4.2. (4.24) u ω u,ω, u L 2 ω (I) H ω(i), and (4.25) u ω0 u,ω, u L 2 ω 0 (I) H ω(i). Proof. We first prove (4.24). The case (iv) B (i.e., ω = ω /2, /2 = ω) is well-known (cf. [3]). For the case (i) B (i.e., ω = ω 5/2, 5/2 ), we recall the Hardy inequality ( ) 2( (4.26) ψ(y)dy x) d dx 4 ψ 2 (x)( x) d dx, x d and (4.27) 0 0 x ( x ) 2( ψ(y)dy + x) d dx 4 0 ψ 2 (x)( + x) d dx, + x d which hold for any measurable function φ(x), and real number d <. Taking ψ = x u and d = 2 (4.28) 0 in (4.26) leads to u 2 (x)( x 2 ) 5/2 dx 0 ( x u) 2 ( x 2 ) /2 dx. 0 u 2 (x)( x) 5/2 0 0 ( x u) 2 ( x) /2 dx A similar inequality holds on the subinterval [, 0] by using (4.27). A combination of them yields u 2 ω x u 2 ω u 2,ω. Since the proofs for the cases (ii) B and (iii) B are essentially the same, we will only consider the case (iii) B. Thanks to (4.28), we have (4.29) On the other hand, 0 0 u 2 (x)( x) 5/2 ( + x) /2 dx 0 ( x u) 2 ( x) /2 dx u 2 (x)( x) 5/2 ( + x) /2 dx A combination of them leads to the desired result The inequality (4.25) can be proved in a similar fashion. u 2 (x)( x) 5/2 ( x u) 2 ( x 2 ) /2 dx. u 2 (x)( x 2 ) /2 dx.

19 HYPERBOLIC EQUATIOS 9 As in the Legendre case, we need to derive some Chebyshev approximation results in suitable weighted Sobolev spaces. Let us define the subspaces of H ω(i) : 0H ω(i) = { u H ω(i) : u( ) = 0 }, 0 H ω(i) = { u H ω(i) : u() = 0 }, and H0,ω (I) = 0Hω(I) 0 Hω(I). Below, we shall use V to denote H0,ω (I), 0 Hω(I) and Hω(I) for the cases (i) B, (ii) B, (iii) B and (iv) B, respectively. 0H ω(i), Define the orthogonal projection: π,ω : V V (defined in (2.2) for each case) by ( (4.30) x (π,ωu u), x v )ω + ( π,ωu ) u, v ω = 0, v V. Lemma 4.3. For any u V B 3/2, 3/2 r (I) with integer r, we have that for each case of (i) B -(iv) B, (4.3) π,ωu u µ,ω µ r xu r ω r 3/2,r 3/2, µ = 0,. If, in addition, u L 2 χ(i) with χ = ω 0 or ω given in (4.23), then (4.32) π,ωu u χ r xu r ω r 3/2,r 3/2. Proof. The estimate (4.3) for (i) B H 0,ω orthogonal projection, and (iv) H ω orthogonal projection, can be found, for instance in [3], an improvement with the semi-norm in the upper bound can be found in [4]. The other two cases of (4.3) are stated in Theorem 3.2 of [4]. Then, a combination of Lemma 4.24 and (4.3) implies (4.32) Convergence of (2.3) with ω = ( x 2 ) /2. Theorem 4.3. Let u (0) = u 0, = π,ω u 0. If u L 2 (0, T ; L 2 ω (I)) L (0, T ; B 3/2, 3/2 r (I)) and t u L 2 (0, T ; B r 3/2, 3/2 (I)) with integer r 2, then (4.33) u u L (0,T ;L 2 ω 0 (I)) + λ u u L 2 (0,T ;L 2 ω (I)) r( t r x u L 2 (0,T ;L 2 ω r 5/2,r 5/2 (I)) + r xu L (0,T ;L 2 ω r 3/2,r 3/2 (I)) ). where λ, ω 0 and ω are the same as in Theorem 2. (also see (4.23)). Proof. Set ê = π,ω u u and e = u u = (u π,ω u) + ê. By an argument similar to the derivation of (4.2), we have (4.34) ê L (0,T ;L 2 ω 0 (I)) + λ ê L 2 (0,T ;L 2 ω (I)) W + W 2 where W = t (π,ω u u) L 2 (0,T ;L 2 ω 2 (I)), and W 2 = x (a(π,ω u u)) + b(π,ω u u) L 2 (0,T ;L 2 ω 2 (I)), ω 2 = ω /2, /2, ω 3/2, /2, ω /2,3/2, ω 3/2,3/2, for the cases (i) B -(iv) B, respectively. Since ω 2 ω, we derive from (4.3) that W r t x r u L 2 (0,T ;L 2 (I)), r 2, ωr 5/2,r 5/2

20 20 JIE SHE & LI-LIA WAG and W 2 r xu r L 2 (0,T ;L 2 (I)), r. ωr 3/2,r 3/2 On the other hand, by (4.32), we have that (4.35) π,ωu u L (0,T ;L 2 ω 0 (I)) + λ π,ωu u L 2 (0,T ;L 2 ω (I)) r( r xu L (0,T ;L 2 ω r 3/2,r 3/2 (I)) + r xu L 2 (0,T ;L 2 ω r 3/2,r 3/2 (I)) ) r xu r L (0,T ;L 2 (I)). ωr 3/2,r 3/2 Hence, using a triangle inequality and the above estimates leads to the desired result. 5. Concluding Remarks We presented in this paper a Legendre and Chebyshev dual-petrov-galerkin method for hyperbolic equations. The dual-petrov-galerkin method is based on a natural variational formulation for hyperbolic equations. A distinctive feature of this variational formulation is that the associated bilinear form for general hyperbolic equations is coercive. An immediate consequence of this property is that the dual-petrov-galerkin method is always stable without any restriction on the coefficients. Another consequence is that one can build robust preconditioners as in the elliptic equations. In fact, by working in the frequency space, we were able to build an optimal (in the sense that for a given accuracy, the required iteration number is independent of the number of modes) preconditioner, which is the sparse matrix associated with an equation with suitable constant coefficients, for the linear system of an implicit time discretization of general hyperbolic equations. This paper is our first effort in developing robust spectral algorithms for hyperbolic equations/systems. In future works, we shall investigate whether the dual-petrov- Galerkin framework can be extended to effectively handle hyperbolic systems, and whether one can build more robust preconditioners using a suitable finite element approximation of the dual-petrov-galerkin formulation. We shall also investigate the numerical and theoretical issues of the dual-petrov-galerkin method for nonlinear hyperbolic equations. Appendix A. Properties of Jacobi polynomials We collect below some relevant formulas of the Jacobi polynomials used in this paper. The Jacobi polynomials Jn α,β (x) (α, β > ) are orthogonal with respect to the Jacobi weight ω α,β = ( x) α ( + x) β, i.e., (A.) Jn α,β (x)jm α,β (x)ω α,β (x)dx = γn α,β δ mn, with (A.2) γ α,β n = I 2 α+β+ Γ(n + α + )Γ(n + β + ) (2n + α + β + )Γ(n + )Γ(n + α + β + ).

21 HYPERBOLIC EQUATIOS 2 There hold the following recursive formulas (cf. Szegö [23] and Askey []): (A.3) (A.4) (A.5) (A.6) J α,β n (x) = J α,β n α,β (x) = Jn (x) Jn α,β (x), α, β > 0; n + α + β ( x)j α+,β n (x) = ( + x)j α,β+ n (x) = α,β {(n + β)jn (x) + (n + α)jn α,β (x)}, α, β > 0; 2 2n + α + β n + α + β + 2 α,β {(n + α + )Jn (x) (n + )J α,β n+ (x)}; α,β {(n + β + )Jn (x) + (n + )J α,β n+ (x)}; (A.7) x Jn α,β (x) = α+,β+ (n + α + β + )Jn (x), n. 2 The Legendre polynomials: L n (x) := Jn 0,0 (x), n 0, satisfy (A.8) (2n + )L n (x) = x L n+ (x) x L n (x), n ; (A.9) ( x 2 n(n + ) ) x L n (x) = 2n + (L n (x) L n+ (x)), n. The Chebyshev polynomials are defined by (A.0) We have that T n (x) = J n 2, 2 J 2, 2 n () (x) = cos(n arccos(x)), n 0. (A.) 2xT n (x) = T n (x) + T n+ (x), n ; (A.2) ( x 2 ) x T n (x) = n 2 (T n (x) T n+ (x)), n. As a consequence, (A.3) ( x) x ( ( + x)tn (x) ) = n 2 T n (x) + T n (x) n + T n+ (x). 2 Appendix B. The proofs of (4.2)-(4.4) and (4.7)-(4.9). Using (A.7), (A.8) and (A.9), we obtain (4.2). Then, the first result of (4.3) comes directly from (A.6). ext, we derive from (A.3), (A.4), (A.7) and (4.) that x φ n (x) (A.7) =, (n + 2)Jn (x) +, (n + )Jn 2 2 (x) (A.3) =, (n + 2)Jn (x) +,0 (n + )(J 2 2 = ( ) (n + 2)Jn, (x) (n + )Jn 0, (x) 2 (A.4) Similarly, we can prove (4.4). = (n + )J,0 (x). n n (x) Jn 0, (x)) + 2,0 (n + )Jn (x)

22 22 JIE SHE & LI-LIA WAG Using (A.7), (4.2) and an induction argument, we find that (B.) xϕ l n (x) = (2n + 3) x l L n+ (x) = κ n,l J l,l n l+2 (x) with κ n,l = (2n + 3)Γ(n + l + ) 2 l. Γ(n + 2) Therefore, (4.7) follows from the orthogonality (A.) with Similarly, we can prove (4.8) and (4.9). µ () n,l = Γ(n + l + ) κ2 n,l γl,l n l+2 = 2(2n + 3) (Γ(n l + 3)). References [] Richard Askey. Orthogonal polynomials and special functions. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 975. [2] Ivo Babuška and Benqi Guo. Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions. umer. Math., 85(2):29 255, [3] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods in Fluid Dynamics. Springer-Verlag, 987. [4] C. Canuto and A. Quarteroni. Error estimates for spectral and pseudospectral approximations of hyperbolic equations. SIAM J. umer. Anal., 9(3): , 982. [5] C. Canuto and A. Quarteroni. Preconditioner minimal residual methods for Chebyshev spectral calculations. J. Comput. Phys., 60:35 337, 985. [6] R. Courant and D. Hilbert. Methods of Mathematical Physics, volume. Interscience Publishers, 953. [7] D. Funaro. Polynomial Approxiamtions of Differential Equations. Springer-verlag, 992. [8] Jonathan Goodman, Thomas Hou, and Eitan Tadmor. On the stability of the unsmoothed Fourier method for hyperbolic equations. umer. Math., 67():93 29, 994. [9] D. Gottlieb and J. S. Hesthaven. Spectral methods for hyperbolic problems. J. Comput. Appl. Math., 28(-2):83 3, 200. umerical analysis 2000, Vol. VII, Partial differential equations. [0] D. Gottlieb and S. A. Orszag. umerical Analysis of Spectral Methods: Theory and Applications. SIAM-CBMS, Philadelphia, 977. [] David Gottlieb. The stability of pseudospectral-chebyshev methods. Math. Comp., 36(53):07 8, 98. [2] David Gottlieb and Eitan Tadmor. The CFL condition for spectral approximations to hyperbolic initial-boundary value problems. Math. Comp., 56(94): , 99. [3] Olivier Goubet and Jie Shen. On the dual Petrov-Galerkin formulation of the KDV equation in a finite interval. Submitted, [4] Ben-yu Guo and Li-lian Wang. Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory, 28(): 4, [5] Benyu Guo, Jie Shen, and Li-Lian Wang. Optimal spectral-galerkin methods using generalized jacobi polynomials. J. Sci. Comput., [6] Z. Jackiewicz and B. D. Welfert. Stability of Gauss-Radau pseudospectral approximations of the one-dimensional wave equation. J. Sci. Comput., 8(2):287 33, [7] Heinz-Otto Kreiss and Joseph Oliger. Comparison of accurate methods for the integration of hyperbolic equations. Tellus, 24:99 25, 972. [8] S. A. Orszag. Spectral methods for complex geometries. J. Comput. Phys., 37:70 92, 980. [9] Seymour V. Parter. Preconditioning Legendre spectral collocation methods for elliptic problems. II. Finite element operators. SIAM J. umer. Anal., 39(): (electronic), 200. [20] Jie Shen. Efficient spectral-galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput., 5: , 994. [2] Jie Shen. Efficient spectral-galerkin method II. direct solvers for second- and fourth-order equations by using Chebyshev polynomials. SIAM J. Sci. Comput., 6:74 87, 995.

23 HYPERBOLIC EQUATIOS 23 [22] Jie Shen. A new dual-petrov-galerkin method for third and higher odd-order differential equations: application to the KDV equation. SIAM J. umer. Anal., 4:595 69, [23] G. Szegö. Orthogonal Polynomials (fourth edition), volume 23. AMS Coll. Publ., 975.