Spectral Collocation Methods for Differential-Algebraic Equations with Arbitrary Index

Transcription

1 J Sci Comput (014) 58: DOI /s Spectral Collocation Methods for Differential-Algebraic Equations with Arbitrary Index Can Huang Zhimin Zhang Received: 3 December 01 / Revised: 7 May 013 / Accepted: 10 July 013 / Published online: 0 July 013 Springer Science+Business Media New York 013 Abstract In this paper, a symmetric Jacobi Gauss collocation scheme is explored for both linear and nonlinear differential-algebraic equations (DAEs) of arbitrary index After standard index reduction techniques, a type of Jacobi Gauss collocation scheme with N knots is applied to differential part whereas another type of Jacobi Gauss collocation scheme with N + 1 knots is applied to algebraic part of the equation Convergence analysis for linear DAEs is performed based upon Lebesgue constant of Lagrange interpolation and orthogonal approximation In particular, the scheme for nonlinear DAEs can be applied to Hamiltonian systems Numerical results are performed to demonstrate the effectiveness of the proposed method Keywords systems Spectral collocation method Differential-algebraic equation Hamiltonian Mathematics Subject Classification (000) Primary: 65L10 Secondary: 60N0 1 Introduction In this paper, we consider the numerical solution of linear differential-algebraic equations (DAEs) with form C Huang (B) Department of Mathematics, Michigan State University, East Lansing, MI 4884, USA canhuang007@gmailcom Z Zhang Department of Mathematics, Wayne State University, Detroit, MI 480, USA ag7761@wayneedu Z Zhang Beijing Computational Science Research Center, Beijing , China

2 500 J Sci Comput (014) 58: { E(t)ẋ(t) = A(t)x(t) + f (t) fall all t I, Bx(t) + Dx(t) = r, (1) where I =[t, t], E, A : I R n n and f : I R n are sufficiently smooth and B, D R d n, d n is the number of differential equations We also consider nonlinear DAEs of the form { F(t, x, ẋ) = 0, () f (x(t), x(t)) = 0, where F and f are smooth functions We assume that these equations possesses a unique solution x Collocation methods for linear DAEs has been extensively studied, see Radau collocation [15], projected piecewise polynomial collocation [1], perturbed collocation [3] and symmetric collocation [9] Collocation methods for general nonlinear DAE is explored in [10] In [9 11], by index reduction technique, one can separate algebraic equations from differential ones Furthermore, an h-version Legendre Gauss scheme for differential part with N knots and h-version Legendre Gauss Lobatto scheme for algebraic part with N + 1 knots are applied to obtain a symmetric method which guarantees consistent approximations at the mesh points In this work, however, we apply a p-version Jacobi Gauss scheme (for differential part) and Jacobi Gauss Lobatto scheme (for algebraic part) to solve linear or nonlinear DAEs Based upon the scheme, a convergence result is established for linear DAEs under standard assumptions and is confirmed by our numerical examples Furthermore, the scheme for nonlinear DAEs can be applied for numerical solution of Hamiltonian systems We compare phase plot of some Hamiltonian systems as DAEs with those in [7], in which a spectral collocation method is designed for Hamiltonian systems as ODEs This paper is organized as follows: In Sect, some preliminary knowledge on the theory of DAEsandJacobipolynomials aregiven In Sect3, we formulate our algorithm for linear DAEs and provide a convergence analysis Two numerical experiments are also included in this section Sect 4 is devoted to the algorithm and numerical experiments for nonlinear DAEs We then give some conclusions in Sect 5 Throughout the paper, C stands for a generic constant that is independent of the number of collocation points N but may depend on the length of time interval t t and the dimension n Preliminaries Application of index reduction techniques on the linear DAEs (1) and nonlinear DAEs (), one obtain with Ê(t)ẋ(t) = Â(t)x(t) + ˆ f (t), t [t, t] (3) Ê =[Ê 1, 0] T, Â =[Â 1, Â ] T, f ˆ =[ˆ f 1, fˆ ] T (4) and { ˆF 1 (t, x, ẋ) = 0, ˆF (t, x) = 0, t [t, t] (5)

3 J Sci Comput (014) 58: Since solutions of (1) and(3), () and(5) are equivalent respectively, we assume that both the linear and nonlinear DAE that we work on is of the index reduced form and without causing any confusion, we keep the same notation as those in (1) and() We denote a and d the size of algebraic and di f f erential part of (3) and(5) Interested readers are referred to [9,10] for more details For the linear DAEs, we have Lemma 1 ([9]) For E, A C k (I, R n n ), there exist point-wise nonsingular P C k 1 (I, R n n ), Q C k (I, R n n ) such that [ ] [ ] Id PEQ =, PAQ PE Q = (6) I a In particular, P has the special structure [ ] P11 P P = 1, with P 0 P 11 (t) R d d, P 1 R d a, P (t) R a a (7) In addition, if f C k 1 (I, R n ),thenx C k 1 (I, R n ) for every solution x of (1) Next, we introduce Jacobi polynomials P n (α,β) the paper It is well-known that P n (α,β) (x), which plays a center role throughout (x) is a class of orthogonal polynomials associate with ( k + α ), k weight ω(x) = (1 x) α (1+x) β,α,β > 1 Under the normalization P (α,β) k (1) = one has the expression, P (α,β) k (x) = 1 k k l=0 ( ) k + α ( ) k + βl (x 1) l (x + 1) k l k l Jacobi polynomials satisfy the three-term recursive relations: where P (α,β) 0 (x) = 1, P (α,β) 1 (x) = 1 [(α β) + (α + β + )x], a 1,k P (α,β) k+1 (x) = a,k P (α,β) k (x) a 3,k P (α,β) k 1 (x), a 1,k = (k + 1)(k + α + β + 1)(k + α + β), a,k = (k + α + β + 1) ( α β ) +xɣ(k + α + β + 3)/ Ɣ(k + α + β), a 3,k = (k + α)(k + β)(k + α + β + ) Especially, Legendre polynomial is exactly a class of Jacobi polynomial with α = β = 0 and Chebyshev polynomials of the first kind is another class with α = β = 1/ uptoa constant It is clear that P n (α,β) (x) forms a complete orthogonal system associate with the norm v ω α,β = ( 1 1 v(x) ω α,β (x)dx) 1/

4 50 J Sci Comput (014) 58: Denote ω(x) = ω 1/, 1/ (x) the Chebyshev weight function and introduce a semi-norm for the simplicity of analysis ( m v H m,n ω ( 1,1) = ) 1/ x k v L ω ( 1,1) k=min(m,n+1) Let IN c denote the interpolation operator on Chebyshev points, then we have [4] u IN c u CN 1/ m u H m,n ω ( 1,1) (8) Let h(t) C k (I, R n ) and write u(x) = h [ t t t+t ] (α,β) (1 + x) +, x [ 1, 1] LetP N be the Jacobi polynomial with degree N and I N u P N [ 1, 1] interpolate u at N + 1 Jacobi Gauss points, Jacobi Gauss Radau points or Jacobi Gauss Lobatto points, then we have the following lemma on the Legesgue constant, see [1] Lemma Let {l j (x)} N j=0 be the Lagrange interpolation polynomials at Jacobi Gauss points, then N I α,β N := max l j (x) x [ 1,1] j=0 { O(log N), 1 <α,β 1 =, O(N γ +1/ (9) ), γ = max(α, β), otherwise The remainder of the interpolation is u(x) I N u(x) = u[x 0, x 1,,x N, x]ν(x), where ν(x) = (x x 0 )(x x 1 )(x x N ) and u[x 0, x 1,,x N, x] is the divided difference of u Hence, u(x) I N u(x) = u[x 0, x 1,,x N, x] P (α,β) N+1 L (x), N+1 Ɣ(N+α+β+3) where L N+1 = N+1 Ɣ(α+β+N+)(N+1)! α+β N+1 is the leading coefficient of P (α,β) π N N+1 (x) Moreover, if u is sufficiently smooth, then the divided difference u[x 0, x 1,,x N, x] = u(n+1) (ξ x ) (N + 1)!, ξ x ( 1, 1), which implies N + 1 u(x) I N u(x) L C N+1 (N + 1)! u(n+1) (ξ x ) L (10) For the sake of analysis, we define the Lagrange interpolation at (N + 1) interpolation points, ie, I N u(x i ) = N u(x i )l i (x), (11) where l i (x) is the Lagrange interpolation basis function at these points It is obvious that for all t [t, t] and x [ 1, 1] i=0 h(t) = u(x), h N (t) = u N (x),

5 J Sci Comput (014) 58: Therefore, (h h N )(t) = (u u N )(x) := e(x) (1) 3 Linear Differential-Algebraic Equations This section is devoted to the symmetric Jacobi collocation methods for linear DAEs and its convergence analysis Firstly, we transform the problem (3) onto an equivalent problem on interval [ 1, 1], b a E(ξ)ẋ(ξ) = A 1(ξ)x(ξ) + f 1 (ξ) 0 = A (ξ)x(ξ) + f (ξ), ξ [ 1, 1] (13) Here, for simplicity of notation, the hats are omitted and subindex of E is suppressed 31 Formulation of Algorithm On the interval [ 1, 1],letρ j s be the zeros of Jacobi polynomial P (α,β) N (x),andσ i s be the associated Jacobi Gauss Lobatto points 1 ρ 0 < <ρ N 1 1, 1 = σ 0 < <σ N = 1 (14) Then, we have the interlacing property [6,13], 1 = σ 0 <ρ 0 <σ 1 < <ρ N 1 <σ N = 1 (15) In order to fix the idea, we use T k, the Chebyshev polynomial of the first kind in algorithms Nevertheless, our method is applicable to all Jacobi polynomials Approximate the true solution of (13)by N xn m (ξ) = Ck m T k(ξ), m = 1,,n (16) k=0 The Jacobi coefficients C m k R(N+1) 1 is determined by the following set of conditions: r = Bx N ( 1) + Dx N (1), b a E(ρ i )ẋ N (ρ i ) = A 1 (ρ i )x N (ρ i ) + f 1 (ρ i ), i = 0,,N 1, 0 = A (σ j )x N (σ j ) + f (σ j ), j = 0,,N (17) Note that the above system has exactly (N +1)n unknowns and d+nd+(n +1)a = (N +1)n equations By the fact that T k (ξ) = ku k 1(ξ), whereu k (ξ) is the Chebyshev polynomial of the second kind and denoting for all i = 1,,n, j = 1,,a and k = 1,,d W 1 = [ 1, 1,,( 1) N ], W =[1, 1,,1 N ], f 1,k = [ f 1,k (ρ 0 ),, f 1,k (ρ N 1 ) ], f, j = [ f, j (σ 0 ),, f, j (σ N ) ], f 1 = [ f 1,1,, f 1,d ] T, f = [ f,1,, f,a ] T

6 504 J Sci Comput (014) 58: NU N 1 (ρ 0 ) 01 NU N 1 (ρ 1 ) U =, 01 NU N 1 (ρ N 1 ) 1 ρ 0 T N (ρ 0 ) 1 σ 0 T N (σ 0 ) 1 ρ 1 T N (ρ 1 ) T =, 1 σ 1 T N (σ 1 ) T =, (18) 1 ρ N 1 T N (ρ N 1 ) 1 σ N T N (σ N ) we have a system AC = F,where C = [ C1 1 ] T,,C1 N,,Cn 1,,Cn N, B W 1 + D W b a E 11U A 1,11 T b a E r 1nU A 1,1n T f 1 A = b a E d1u A 1,d1 T b a E dnu A 1,dn T, F =, (19) A,11 T A,1n T f A,a1 T A,an T and denotes the Kronecker product Once C is determined, we plug it back to (16) to obtain x N 3 Convergence Results WetrytoderiveanerrorestimateintheL -norm of the collocation scheme (17) Since N-point Jacobi Gauss Lobatto points with parameters (α, β) are zeros of P (α+1,β+1) N (t) (±1 excluded) [14], throughout the paper we only need to consider the case when both ρ i s and σ j s are zeros of certain type of Jacobi polynomials Moreover, it is well-known that both ρ j s and σ j s are dense in [ 1, 1] as N Note that the distance between any two consecutive zeros of any type of Jacobi polynomials is bound by 1 t N, t [ 1, 1] up to a constant independent of N and t [8] We derive that ρ i σ i = O(N 1 ), i = 1,,N 1 To obtain the solvability of our algorithm, let us rewrite our algorithm in its equivalent form For each component of x m, its approximation N xn m = Ck m T k(ξ) = k=0 N xk,m N l k(ξ), (0) where l k (ξ) is the Lagrange interpolation polynomial on point set σ j s Denote v ki = l i (ρ k), u ki = l i (ρ k ), E i = E(ρ i ), A 1i = A 1 (ρ i ), A j = A (σ j ), and x N = k=0 [ x N 0,1, x N 0,,,x N 0,n,,x N N,1,,x N N,n] T,

7 J Sci Comput (014) 58: for k = 1,,N, i = 1,,N Plugging (0)into(13) yields a system A x N = F,where v 00 b a E v 0 u 00 A 10 0N b a E 0 u 0N A 10 A 0 0 v A N 1,0 v = b a E N 1 N 1,N 1 v b a E N 1,N N 1 b a E N 1 u N 1,0 A 1,N 1 u N 1,N 1 A 1,N 1 u N 1,N A 1,N 1 (1) 0 A,N 1 0 C D A (1) F = [f 1 (ρ 0 ) f (σ 0 ) f 1 (ρ N 1 ) f (σ N 1 ) r f (1)] T Here, a missing condition A (1)x N (1) = f (1) is used Hence, if A is regular, our collocation algorithm is stable To reorder rows and columns, we define U N = I d I a 0 I d I a 0 I d 0 0 I a I d 0 0 I a R (N+1)n (N+1)n () Remark 1 As is well-known, Lagrange interpolation is not stable for large N Moreover, it requires a large computational cost to find the base functions Hence, the equivalent algorithm is exhibited only for a theoretical purpose Lemma 31 For N sufficiently large, { O(N l j (ρ i ) = 1 ), i = j and i = j 1, O(1), otherwise Proof Let δ be a fixed number such that δ<1 ρ i By the interlacing property, we can choose δ so small that B(σ j,δ) ρ ={ρ j 1,ρ j }From[15] (pp 336), if σ j ρ i >δ, otherwise, l j (ρ i ) = O ( N 1/) P (α,β) N+1 (σ j ) 1, = O ( N 1/)( j α+3/ N α + j β+3/ N β ), O ( N 1), (3) l j (ρ i ) = O ( N ( ) (α,β) P(α,β) 1/) N+1 σ j P N+1 (ρ i ) = O(1) (4) σ j ρ i As an illustration, we plot an l 1 (x) for N = 5 and (α, β) = (1/, /3),seeFig1Itis clear that l 1 (ρ j ) has the largest modulus at ρ 11 and ρ 1

8 506 J Sci Comput (014) 58: l 1 (ρ i ) σ j l 1 ([ 1,1]) Fig 1 Aplotofl 1 (x) for N = 5 and (α, β) = (1/, /3) Next, we prove the invertibility of A by multiplying it from the left and from the right, respectively, by ( ( ) [ ] ) P11 (ρ T P = diag diag i ) P 1 (σ i ) I 0,, 0 P (σ i ) 0 P i=0,,n 1 (1) ( ) T Q = diag Q(σ 0 ),,Q(σ N 1 ), Q(1) (5) Lemma 3 Assume A, Q C ThenA is invertible for sufficiently large N Proof Since l i (x) = P N+1 (x) P N+1 (σ, after a simple calculation, we have i )(x σ i ) v ij = l i (ρ j ) = l i (ρ j ) ρ j σ i (6) Clearly, from Lemma (31), O(N), j = i, or i 1; v ij = O(1), j = i, or i 1and σ j ρ i =O(N 1 ); o(1), otherwise (7) Hence, for j = i or i 1, I 1 =[P 11 (ρ i ) P 1 (σ i )] [ ] vij b a E i u ij A 1i Q(σ j ) (8) 0 = v ij b a (P 11E)(ρ i )Q(σ j ) u ij (P 11 A 1 )(ρ i )Q(σ j ) = v ij b a (P 11EQ)(ρ i ) u ij (P 11 A 1 )(ρ i )Q(σ j ) + e 1,

9 J Sci Comput (014) 58: where e 1 = v ij b a (P 11E)(ρ i )(Q(σ j ) Q(ρ i )) Apply the smoothness assumption of P, E and Q, e 1 = l i (ρ j ) b a (P 11E)(ρ i ) Q(ξ)(ρ j σ i ), = l i (ρ j )(P 11 E)(ρ i ) Q(ξ), = O(1)l i (ρ j ) (9) However, in this case, l i (ρ j )/l i (ρ j ) 0 by the L Hopital s rule Therefore, (9) andthe definition of u ij yield Similarly, if j = i 1, I 1 = v ij b a [I 0]+O(1) If j = i, [ ] vij I =[P 11 (ρ i ) P 1 (σ i )] b a E i u ij A 1i Q(σ i ) A i I 1 = v ij b a [I 0]+o(v ij) (30) At last, = v ii b a (P 11E)(ρ i )Q(σ i ) u ii (P 11 A 1 )(ρ i )Q(σ i ) (P 1 A Q)(σ i ) = v ii b a (P 11EQ)(ρ i ) u ii (P 11 A 1 )(ρ i )Q(σ i ) (P 1 A Q)(σ i ) + e, = v ii [I 0]+O(1) (31) b a [ ][ I 0 B 0 P (1) 0 [ ][ I 0 0 P (1) ] Q(1) = D A (1) [ B11 B ] Q(1) = ], [ D11 D 1 0 I ] Hence, we have M = T P A T Q = v 00 b a [I v d d 0]+O(1) 0,N 1 b a [I d d 0]+O( N 1 ) v 0N b a [I d d 0]+O( N 1 ) I a a 0 0 v N 1,0 b a [I d d 0]+O( N 1 ) v N 1,N 1 b a [I d d 0]+O(1) v N 1,N b a [I d d 0]+O(1) 0 I a a 0 B 11 B 1 D 11 D I a a (3) Rewrite the above matrix as [ ] Ã B M = C D

10 508 J Sci Comput (014) 58: Clearly, if D11 1 exists, [ D 1 D 1 = Note that [ ][ ] à B I 0 C D D 1 = C I 11 D 1 11 D 1 0 I ] [ à B D 1 ] C B 0 D Therefore, the invertibility of à B D 1 C implies the invertibility of A By a simple matrix manipulation, à B D 1 C = v 00 b a [I d d 0] v 0N b a D 1 11 B v 11 + O(1) 0,N 1 b a [I d d 0]+O( N 1 ) I a a 0 v N 1,0 b a [I d d 0] v N 1,N b a D11 1 B 11 + O(1) v N 1,N 1 b a [I d d 0]+O(1) 0 I a a It is clear that the matrix is diagonal dominant, hence invertible This result indicates that our algorithm is well-posed In the following, we find the error estimates for the algorithm Theorem 33 Let x(ξ) be the true solution of transformed DAE (13) and x N (ξ) be the approximation given by (17) based upon our collocation method If the given data E(ξ), A(ξ) C m+1 ([ 1, 1], R n n ), and f C m ([ 1, 1], R n ), then for sufficiently large N, { (log N)N x x N C 3/ m, 1 <α,β< 1/, N γ + m (33), γ = max(α, β), otherwise Proof Let I N be the interpolation operator on σ 0,,σ N and Ĩ N 1 be the one on ρ 0,,ρ N 1 Clearly, the true solution satisfies r = Bx( 1) + Dx(1), b a E(ρ i )ẋ(ρ i ) = A 1 (ρ i )x(ρ i ) + f 1 (ρ i ), i = 0,,N 1, 0 = A (σ j )x(σ j ) + f (σ j ), j = 0,,N (34) Multiplying both sides of the first and the third equation of (34)byl j (ξ), the second equation by l i (ξ), respectively, and summing over i, j yield r = BI N x( 1) + DI N x(1), b a ĨN 1(Eẋ) = Ĩ N 1 (A 1 x) + Ĩ N 1 f 1, 0 = I N (A x) + I N f (35) We follow the same fashion for (17) to obtain a system for x N, then subtract the system obtained from (35), 0 = BI N (x( 1) x N ( 1)) + DI N (x(1) x N (1)), 0 = b a ĨN 1(E 1 ẋ Eẋ N ) Ĩ N 1 (A 1 x A 1 x N ), 0 = I N (A x A x N ) (36)

11 J Sci Comput (014) 58: Recall the definition of I c N SinceI N A and I N x are polynomials, We have the following estimate from (558) of [4], I N (A x) (I N A )(I N x) = IN c (A x) (IN c A )(IN c x) = IN c ((I N c A )(IN c x)) (I N c A )(IN c x) CN 1/ m IN c A IN c x Hω m,n ( 1,1) (37) Similar estimation can be obtained for I N (A x N ) (I N A )(I N x N ), Ĩ N 1 (Eẋ) (Ĩ N 1 E)(Ĩ N 1 ẋ), Ĩ N 1 (Ex N ) (Ĩ N 1 E)(Ĩ N 1 x N ), Ĩ N 1 (A 1 x) (Ĩ N 1 A 1 ) (Ĩ N 1 x) and Ĩ N 1 (A 1 x N ) (Ĩ N 1 A 1 )(Ĩ N 1 x N ) However, all these terms are highorder terms Combine all these estimates, (36) can be written as 0 = BI N (x( 1) x N ( 1)) + DI N (x(1) x N (1)), O(N 1/ m ) = b a (I N 1E 1 )[Ĩ N 1 ẋ Ĩ N 1 ẋ N ] (Ĩ N 1 A 1 )[Ĩ N 1 x Ĩ N 1 x N ], O(N 1/ m ) = (I N A )(I N x I N x N ) (38) Also, by simple calculation, Ĩ N 1 ẋ Ĩ N 1 ẋ N (I N x) + (I N x N ) = Ĩ N 1 ẋ (I N x) CN 3/ m (39) We obtain 0 = BI N (x( 1) x N ( 1)) + DI N (x(1) x N (1)), O(N 3/ m I N ) = b a (I N E 1 )[(I N x) ẋ N ] (I N A 1 )[I N x x N ], O(N 1/ m ) = (I N A )(I N x x N ) Writing the right hand side of the last two groups of equations as w and applying the transformation in Lemma 1 yields that [ ] [ ][ ] [ ] Id 0 ][ẏ1 0 0 y1 g1 = + (40) 0 0 ẏ 0 I a y g [ ] [ ] y1 ( 1) y1 (1) [B 1 B ] +[D y ( 1) 1 D ] = 0, (41) y (1) where [ ] [ ] y1 g1 I N x x N = Q, Pw =, BQ( 1) =[B 1 B ], DQ(1) =[D 1, D ] y g Matrix functions P and Q are given in Lemma 1 The differences P(E I N E)Q = O(N m 1 ) and P(A I N A)Q = O(N m 1 ) are absorbed into functions g Clearly, ξ y 1 (ξ) = y 1 ( 1) + 1 g 1 (s)ds, y (ξ) = g (ξ) (4) The equation is equivalent to [11] { ẏ1 = g 1 (ξ), y 1 ( 1) = (B 1 + D 1 ) g 1(s)ds y (ξ) = g (ξ) (43)

12 510 J Sci Comput (014) 58: Hence, it is enough for us to consider the corresponding initial value condition for our problem Obviously, y C g, which implies Therefore, I N x x N C I N N 3/ m (44) x x N x I N x + I N x x N { (log N)N C 3/ m, 1 <α,β< 1/, N γ + m, γ = max(α, β), otherwise (45) 33 Numerical Experiments We present two examples here to illustrate the effectiveness of our algorithm Both examples are from [9] Unconfined to the type of points in our theory, we use three different types of collocation points, which is presented in Table 1 Example 1 Consider the DAE t 0 e t/ 1 t 0 ẋ = x + 0, t [ 5, 0] 1 t 1 0 t 1 0 [1 70] x( 5) +[041] x(0) = 6 This is an index-two problem with solution x(t) = e (1 t/ t, 1 ), t + 4t + 8 Since the solution is smooth, from Table and Fig, we actually observe a super geometric rate of convergence (O ( γ ) σ N N ), where γ,σ are some constants [16] Table 1 Three sets of collocation points ρ σ Set 1 Gauss Legendre points Gauss Legendre Lobatto points Set Gauss Legendre Lobatto Gauss Legendre Radau points points Set 3 Gauss Legendre points Gauss Legendre points Table Errors for Example 1 Number of points Set 1 Set Set e 8836e 59604e e 6 007e e e e e e e e 14

13 J Sci Comput (014) 58: Set Set Set 3 Fig Numerical errors for Example 1 Example Consider the problem of form (1) with 1 t 0 E(t) = t 1 t, p(t) t(p(t) ) 0 κ 1 t t ( t)κ κ 1 A(t) = t t 1 t κ 1 t 1 t + κ κp(t) +t, κt(t 3) p(t) t p(t) t 4κ κ(p(t)( t) t 3 + 6t 4) 3 t t f (t) = + (κ+)p(t)+ṗ(t) tp(t) t 4 (t 4) (p(t) ) t 3 t + t ) e t, (46) where t [0, 1], parameter κ R and a smooth function p is a C 1 smooth function The boundary condition is x 1 (0) = 1 We choose κ = 0 and ( ( )) t 1/3 p(t) = 1 + erf, ε = 10 ε There exists an inner layer at t = 1/3 inp and also in the solution x(t) = et t t t t + tp(t) t 4 1 t t t + p(t) t 4 t +1 t If ε is too small, the solution is not smooth Hence, it is not advantageous to use the p-version method For the choice of ε = 10,Table3 records the numerical errors and Fig 3 shows the rate of convergence as desired

14 51 J Sci Comput (014) 58: Table 3 Errors for Example Number of points Set 1 Set Set e 1090e 19384e e e e e e e e e e Set 1 Fig 3 Numerical errors for Example Set Set 3 4 Nonlinear Differential-Algebraic Equations Similar as linear DAEs, we first transform (5) into its canonical form, F 1 (ξ, x, b a ẋ) = 0, F (ξ, x) = 0, ξ [ 1, 1], f (x( 1), x(1)) = 0 (47) 41 Formulation of Algorithm The collocation points that we use are exactly the same as those in (14), where ρ is for differential part and σ is for algebraic part Denote x m N (ξ) = N k=0 C m k T k(ξ), m = 1,,n the approximation of x in (47) We have the following set of conditions nonlinear equations: f (x N ( 1), x N (1)) = 0, F 1 (ρ i, x N (ρ i ), b a ẋn (ρ i )) = 0, i = 0,,N 1, F (σ j, x N (σ j )) = 0, j = 0,,N (48)

15 J Sci Comput (014) 58: Now we convert it into a nonlinear equation of C = (C 1,,C n ) T ) f ([W 1 C 1,,W 1 C n ] T, [W C 1,,W C n ] T = 0, ) F 1 (ρ,[tc 1,,TC n ] T, b a [UC1,,UC n ] T = 0, ( ) F σ, [ TC 1,, TC n ] T = 0, (49) where T, U and T are defined in (18) We use Newton-iteration to solve the above system 4 Numerical Experiments for Nonlinear DAEs We present four examples in this section to illustrate the application of spectral collocation method for nonlinear DAEs One important field of application is to solve Hamiltonian systems [5] Given a Hamiltonian system: { dp(t) dt = H dq(t) dt q, p(0) = p 0, = p H, q(0) = q 0, where p(t), q(t) R n Setting (x 1,,x n ) = (p, q) T, x n+1 = H, we obtain a nonlinear DAE on the interval [0, T ] dx i (t) = H i = 1,,n (Free) dt q i, dx j (t) = H dt p, j = n + 1,n (Free) dx n+1 (t) dt = 0, H(x 1,,x n, x n+1,,x n ) x n+1 = 0, x i (0) = x 0, i = 1,,n, x n+1 (0) = H 0 It should be pointed out that the system above is overdetermined and in practice, one need to abandon one equation among all free equations Throughout this subsection, we choose ρ as Legendre Gauss points and σ as Legendre Gauss Lobatto points Remark On the interval [nt,(n+1)t], taking the transform t = nt+t + T ξ, ξ [ 1, 1], we can obtain the same system as (51) except that x i ( 1) =ˆx 0, x n+1 ( 1) = Ĥ 0,where ˆx 0 and Ĥ 0 are the value of x i (1) and x n+1 (1) obtained from the preceding interval, respectively To better approximate energy, we may specify initial condition as x i (0) = x 0, i = 1,,n and take a poststablization process [] at the end of each step: where h = H x and F = H T (HH T ) 1 x i ( 1) =ˆx 0, x i ( 1) = x i F( x)h( x), (50) (51) Example 3 We first consider an example from [10] ẋ 1 = (ɛ + x p (t))x 4 +ṗ 1 (t), ẋ = ṗ (t), ẋ 3 = x 4, 0 = (x 1 p 1 (t))(x 4 e t )

16 514 J Sci Comput (014) 58: Table 4 Errors for Example 3 Number of points max x(σ i ) x i 19566e0 6436e e e 15 1 i 4 L norm number of points q p Fig 4 (Left) numerical errors for example 3; (Right) a phase plot of (p, q) on [0, 10 4 ] for Example 4 Choosing the boundary condition x 1 (0) = P 1 (0) + ɛ, x 3 (0) = 1, x (1) = p (1), the exact solution of the problem is ) x (t) = (ɛe t + p 1 (t), p (t), e t, e t We choose p 1 (t) = sin(4πt), p (t) = sin(t), andɛ = 1/ as parameters Using our algorithm, we obtain Table 4 and Fig 4(Left) Clearly, the convergence rate is geometric Example 4 Consider a simple linear Hamiltonian system [7]: { p (t) = 4q(t), q (t) = p(t) with initial condition p(0) = 0, q(0) = 06 and Hamiltonian H(p, q) = 1/p + q = 07 Although the Hamiltonian system itself is linear, if we consider the Hamiltonian as a variable and apply our algorithm, we obtain a nonlinear DAE In [7], the system is treated as ODEs and numerically solved by spectral collocation method On [0, 10 5 ], their algorithm has an energy error at the terminal point whereas our algorithm has an error Furthermore, applying the poststablization approach mentioned above, we can reduce the error to machine epsilon Hence, our algorithm is much more accurate from this point of view Figure 4(Right) shows that our algorithm keeps the symplectic structure of the system also However, our algorithm takes longer CPU time to obtain the result, which is 1, 14 s while theirs takes only 61 s for 0 collocation points

17 J Sci Comput (014) 58: q Prey x 1 Predator x p Fig 5 (Left) a phase plot of (p, q) on [0, 5000] for Example 6; (Right) prey/predator interaction in one cycle for Example 5 Example 5 Consider the Lotka Volterra system for a predator/prey interaction, which consists of two equations [10] { ẋ1 = x 1 (1 x ), (5) ẋ = cx (1 x 1 ), where c > 0 In the system, the quantity, H = c(x 1 log x 1 ) + (x log x ) remains constant along time, which forces solutions x 1 and x to be periodic But the period is unknown As in [10], we introduce new variables x 3 = H and x 4 = T and transform the equation into the interval [0, 1] ẋ 1 = x 1 (1 x )x 4, ẋ 3 = 0, (53) ẋ 4 = 0, c(x 1 log x 1 ) + (x log x ) x 3 = 0, t [0, 1] with initial conditions x 1 (0) = 1, x 1 (0) = x 1 (1), andx 3 (0) = For the choice of c = 1, we obtain T converges to as the number of collocation points increases Figure 5(Right) represents the computed interaction between prey and predator in one cycle for 30 collocation points Example 6 Consider a system with Hamiltonian H(p, q) = p q + q 4 [7] The corresponding system of nonlinear ODEs is { ṗ = q 4q 3, q = p with initial condition p(0) = 0, q(0) = 073 Hence, the Hamiltonian is Our DAE algorithm takes only 17s to obtain a result on [0, 5000], with an energy error It can be further reduced to machine epsilon by applying the poststablization process A phase plot can be found in Fig 5(Left)

18 516 J Sci Comput (014) 58: Conclusion In this work, a Jacobi spectral collocation method is applied to solve both linear and nonlinear DAE with arbitrary index We apply a type of Jacobi Gauss collocation scheme with N knots to differential part of DAEs whereas another type of Jacobi Gauss collocation scheme with N + 1 knots to algebraic part of the equation Convergence analysis for linear DAEs shows that the spectral collocation method works efficiently if the solution itself is smooth enough In particular, the scheme for nonlinear DAEs can be applied to solve Hamiltonian systems and it keeps the symplectic structure of the system However, it may take more CPU seconds to obtain a result compared with the method in [7] since we enhance the dimension of the problems References 1 Ascher, U, Petzold, L: Projected collocation for higher-order higher-index differential-algebraic equations J Comput Appl Math 43, (199) Ascher, U, Petzold, L: Computer methods for ordinary differential equations and differential-algebraic equations SIAM, Philadelphia (1998) 3 Bai, Y: A perturbed collocation method for boundary value problems in differential-algebraic equations Appl Math Comput 45, (1991) 4 Canuto, C, Hussaini, M, Quarteroni, A, Zang, T: Spectral Methods: Fundamentals in Single Domains Springer, Berlin (006) 5 Cherry, T: On periodic solutions of Hamiltonian systems of differential equations Philosophical Transactions of the Royal Society of London Series A, Containing Papers of a Mathematical or Physical Character 7, (198) 6 Cimwanga, NM: Interlacing zeros of linear combinations of classical orthogonal polynomials Thesis, University of Pretoria (009) 7 Kanyamee, N, Zhang, Z: Comparison of a spectral collocation method and symplectic methods for Hamiltonian systems Int J Numer Anal Mod 8, (011) 8 Krasovsky, IV: Asymptotic distribution of zeros of polynomials satisfying difference equations J Comput Appl Math 150, (003) 9 Kunkel, P, Stöver, R: Symmetric collocation methods for linear differential-algebraic boundary problems Numer Math 91, (00) 10 Kunkel, P, Mehrmann, V, Stöver, R: Symmetric collocation for unstructured nonlinear differentialalgebraic equations of arbitrary index Numer Math 98, (004) 11 Kunkel, P, Mehrmann, V: Differential-Algebraic Equations: Analysis and Numerical Solution European Mathematical Society, Zürich (006) 1 Mastroianni, G, Occorsio, D: Optimal systems of nodes for Lagrange interpolation on bounded intervals: a survey J Comput Appl Math 134, (001) 13 Shen, J, Tao, T, Wang, LL: Spectral Methods: Algorithms, Analysis and Applications Springer, New York (011) 14 Stancu, DD, Coman, G, Blaga, P: Numerical Analysis and Approximation theory (in Romanian), vol II Cluj University Press, Cluj-Napoca (00) 15 Szegö, G: Orthogonal Polynomials, vol XXIII American Mathematical Society, Providence (1939) 16 Zhang, Z: Superconvergece of spectral collocation and p-version methods in one dimensional problems Math Comput 74, (005)