Introduction to spectral methods

Transcription

1 Introduction to spectral methods Eric Gourgoulhon Laboratoire de l Univers et de ses Théories (LUTH) CNRS / Observatoire de Paris Meudon, France Based on a collaboration with Silvano Bonazzola, Philippe Grandclément, Jean-Alain Marck & Jérome Novak Eric.Gourgoulhon@obspm.fr 4th EU Network Meeting, Palma de Mallorca, Sept. 2002

2 Plan 1 1. Basic principles 2. Legendre and Chebyshev expansions 3. An illustrative example 4. Spectral methods in numerical relativity

3 2 1 Basic principles

4 Solving a partial differential equation 3 Consider the PDE with boundary condition Lu(x) = s(x), x U IR d (1) Bu(y) = 0, y U, (2) where L and B are linear differential operators. Question: What is a numerical solution of (1)-(2)? Answer: It is a function ū which satisfies (2) and makes the residual R := Lū s small.

5 What do you mean by small? 4 Answer in the framework of Method of Weighted Residuals (MWR): Search for solutions ū in a finite-dimensional sub-space P N of some Hilbert space W (typically a L 2 space). Expansion functions = trial functions : basis of P N : (φ 0,..., φ N ) ū is expanded in terms of the trial functions: ū(x) = N ũ n φ n (x) Test functions : family of functions (χ 0,..., χ N ) to define the smallness of the residual R, by means of the Hilbert space scalar product: n=0 n {0,..., N}, (χ n, R) = 0

6 Various numerical methods 5 Classification according to the trial functions φ n : Finite difference: trial functions = overlapping local polynomials of low order Finite element: trial functions = local smooth functions (polynomial of fixed degree which are non-zero only on subdomains of U) Spectral methods : trial functions = global smooth functions (example: Fourier series)

7 Various spectral methods 6 All spectral method: trial functions (φ n ) = complete family (basis) of smooth global functions Classification according to the test functions χ n : Galerkin method: test functions = trial functions: χ n = φ n and each φ n satisfy the boundary condition : Bφ n (y) = 0 tau method: (Lanczos 1938) test functions = (most of) trial functions: χ n = φ n but the φ n do not satisfy the boundary conditions; the latter are enforced by an additional set of equations. collocation or pseudospectral method: test functions = delta functions at special points, called collocation points: χ n = δ(x x n ).

8 Solving a PDE with a Galerkin method 7 Let us return to Equation (1). Since χ n = φ n, the smallness condition for the residual reads, for all n {0,..., N}, (φ n, R) = 0 (φ n, Lū s) = 0 ( ) N φ n, L ũ k φ k (φ n, s) = 0 N ũ k (φ n, Lφ k ) (φ n, s) = 0 N L nk ũ k = (φ n, s), (3) where L nk denotes the matrix L nk := (φ n, Lφ k ). Solving for the linear system (3) leads to the (N + 1) coefficients ũ k of ū

9 Solving a PDE with a tau method 8 Here again χ n = φ n, but the φ n s do not satisfy the boundary condition: Bφ n (y) 0. Let (g p ) be an orthonormal basis of M + 1 < N + 1 functions on the boundary U and let us expand Bφ n (y) upon it: Bφ n (y) = M b pn g p (y) p=0 The boundary condition (2) then becomes Bu(y) = 0 N M ũ k b pk g p (y) = 0, p=0 hence the M + 1 conditions: N b pk ũ k = 0 0 p M

10 Solving a PDE with a tau method (cont d) 9 The system of linear equations for the N + 1 coefficients ũ n is then taken to be the N M first raws of the Galerkin system (3) plus the M + 1 equations above: N L nk ũ k = (φ n, s) 0 n N M 1 N b pk ũ k = 0 0 p M The solution (ũ k ) of this system gives rise to a function ū = N ũ k φ k such that Lū(x) = s(x) + M τ p φ N M+p (x) p=0

11 Solving a PDE with a pseudospectral (collocation) method 10 This time: χ n (x) = δ(x x n ), where the (x n ) constitute the collocation points. The smallness condition for the residual reads, for all n {0,..., N}, (χ n, R) = 0 (δ(x x n ), R) = 0 R(x n ) = 0 Lu(x n ) = s(x n ) N Lφ k (x n )ũ k = s(x n ) (4) The boundary condition is imposed as in the tau method. One then drops M + 1 raws in the linear system (4) and solve the system N Lφ k (x n ) ũ k = s(x n ) 0 n N M 1 N b pk ũ k = 0 0 p M

12 What choice for the trial functions φ n? 11 Periodic problem : φ n = trigonometric polynomials (Fourier series) Non-periodic problem : φ n = orthogonal polynomials

13 12 2 Legendre and Chebyshev expansions

14 Legendre and Chebyshev polynomials 13 Families of orthogonal polynomials on [ 1, 1] : Legendre polynomials: 1 1 P m (x)p n (x) dx = 2 2n + 1 δ mn Chebyshev polynomials: 1 dx T m (x)t n (x) = 1 x 2 1 π 2 (1 + δ 0n) δ mn [from Fornberg (1998)] P 0 (x) = 1, P 1 (x) = x, P 2 (x) = 3 2 x2 1 2 T 0 (x) = 1, T 1 (x) = x, T 2 (x) = 2x 2 1 Both Legendre and Chebyshev polynomials are a subclass of Jacobi polynomials

15 Properties of Chebyshev polynomials 14 Definition: cos nθ = T n (cos θ) Recurrence relation : T n+1 (x) = 2xT n (x) T n 1 (x) Eigenfunctions of the singular Sturm-Liouville problem: d dx ( 1 x 2 dt ) n = n2 dx T n(x) 1 x 2 Orthogonal family in the Hilbert space L 2 w[ 1, 1], equiped with the weight w(x) = (1 x 2 ) 1/2 : (f, g) := 1 1 f(x) g(x) w(x) dx

16 Polynomial interpolation of functions 15 Given a set of N + 1 nodes (x i ) 0 i N in [ 1, 1], the Lagrangian interpolation of a function u(x) is defined by the N-th degree polynomial: I N u(x) = N u(x i ) i=0 N ( ) x xj j=0 j i x i x j Cauchy theorem: there exists x 0 [ 1, 1] such that u(x) I N u(x) = 1 (N + 1)! u(n+1) (x 0 ) N (x x i ) i=0 Minimize u(x) I N u(x) independently of u minimize N (x x i ) i=0

17 Chebyshev interpolation of functions 16 Note that N (x x i ) is a polynomial of degree N + 1 of the type x N+1 + a N x N + i=0 (leading coefficient = 1). Characterization of Chebyshev polynomials: Among all the polynomials of degree n with leading coefficient 1, the unique polynomial which has the smallest maximum on [ 1, 1] is the n-th Chebyshev polynomial divided by 2 n 1 : T n (x)/2 n 1. = take the nodes x i to be the N + 1 zeros of the Chebyshev polynomial T N+1 (x) : N (x x i ) = 1 2 N T N+1(x) i=0 x i = cos ( ) 2i + 1 2(N + 1) π 0 i N

18 Spectral expansions : continuous (exact) coefficients 17 Case where the trial functions are orthogonal polynomials φ n in L 2 w[ 1, 1] for some weight w(x) (e.g. Legendre (w(x) = 1) or Chebyshev (w(x) = (1 x 2 ) 1/2 ) polynomials). The spectral representation of any function u is its orthogonal projection on the space of polynomials of degree N: P N u(x) = N ũ n φ n (x) n=0 where the coefficients ũ n are given by the scalar product: ũ n = 1 (φ n, φ n ) (φ n, u) with (φ n, u) := 1 1 φ n (x) u(x) w(x) dx (5) The integral (5) cannot be computed exactly...

19 Spectral expansions : discrete coefficients 18 The most precise way of numerically evaluating the integral (5) is given by Gauss integration : 1 N f(x) w(x) dx = w i f(x i ) (6) 1 where the x i s are the N + 1 zeros of the polynomial φ N+1 and the coefficients w i are N 1 the solutions of the linear system x i j w j = x i w(x) dx. j=0 Formula (6) is exact for any polynomial f(x) of degree 2N + 1 Adaptation to include the boundaries of [ 1, 1] : x 0 = 1, x 1,..., x N 1, x N = 1 Gauss-Lobatto integration : x i = zeros of the polynomial P = φ N+1 + λφ N + µφ N 1, with λ and µ such that P ( 1) = P (1) = 0. Exact for any polynomial f(x) of degree 2N 1. i=0 1

20 Spectral expansions : discrete coefficients (con t) 19 Define the discrete coefficients û n to be the Gauss-Lobatto approximations of the integrals (5) giving the ũ n s : û n := 1 (φ n, φ n ) N w i φ n (x i ) u(x i ) (7) i=0 The actual numerical representation of a function u is then the polynomial formed from the discrete coefficients: N I N u(x) := û n φ n (x), n=0 instead of the orthogonal projection P N u involving the ũ n. Note: if (φ n ) = Chebyshev polynomials, the coefficients (û n ) can be computed by means of a FFT [i.e. in N ln N operations instead of the N 2 operations of the matrix product (7)].

21 Aliasing error 20 Proposition: I N u(x) is the interpolating polynomial of u through the N + 1 nodes (x i ) 0 i N of the Gauss-Lobatto quadrature: I N u(x i ) = u(x i ) 0 i N On the contrary the orthogonal projection P N u does not necessarily pass through the points (x i ). The difference between I N u and P N u, i.e. between the coefficients û n and ũ n, is called the aliasing error. It can be seen as a contamination of û n by the high frequencies ũ k with k > N, when performing the Gauss-Lobato integration (7).

22 Illustrating the aliasing error: case of Fourier series 21 Alias of a sin( 2x) wave by a sin(6x) wave Alias of a sin( 2x) wave by a sin( 10x) wave [from Canuto et al. (1998)]

23 Convergence of Legendre and Chebyshev expansions 22 Hyp.: u sufficiently regular so that all derivatives up to some order m 1 exist. Legendre: truncation error : P N u u L 2 C m N m u (k) L 2 interpolation error : P N u u I N u u L 2 C N m 1/2V (u(m) ) C m u (k) L 2 N m 1/2 Chebyshev: truncation error : P N u u L 2 w C N m P N u u interpolation error : I N u u L 2 w C N m I N u u m u (k) L 2 w C(1 + ln N) N m m u (k) m u (k) L 2 w C N m 1/2 m u (k) L 2 w

24 Evanescent error 23 From the above decay rates, we conclude that for a C function, the error in the spectral expansion decays more rapidly than any power of 1/N. In practice, it is an exponential decay. Such a behavior is a key property of spectral methods and is called evanescent error. (Remember that for a finite difference method of order k, the error decays only as 1/N k ).

25 24 3 An example... at last!

26 A simple differential equation with boundary conditions 25 Let us consider the 1-D second-order linear (P)DE with the Dirichlet boundary conditions d 2 u dx 2 4du dx + 4u = ex + C, x [ 1, 1] (8) and where C is a constant: C = 4e/(1 + e 2 ). The exact solution of the system (8)-(9) is u( 1) = 0 and u(1) = 0 (9) u(x) = e x sinh 1 sinh 2 e2x + C 4

27 Resolution by means of a Chebyshev spectral method 26 Let us search for a numerical solution of (8)-(9) by means of the five first Chebyshev polynomials: T 0 (x), T 1 (x), T 2 (x), T 3 (x) and T 4 (x), i.e. we adopt N = 4. Let us first expand the source s(x) = e x + C onto the Chebyshev polynomials: P 4 s(x) = 4 s n T n (x) and I 4 s(x) = 4 ŝ n T n (x) n=0 n=0 with s n = 2 π(1 + δ 0n ) 1 1 dx T n (x)s(x) 1 x 2 and ŝ n = 2 π(1 + δ 0n ) 4 w i T n (x i )s(x i ) i=0 the x i s being the 5 Gauss-Lobatto quadrature points for the weight w(x) = (1 x 2 ) 1/2 : {x i } = { cos(iπ/4), 0 i 4} = { 1, 1 2, 0, } 1, 1 2

28 The source and its Chebyshev interpolant s(x) I 4 s(x) 1 y x ŝ 0 = , ŝ 1 = 1.130, ŝ 2 = , ŝ 3 = , ŝ 4 =

29 Interpolation error and aliasing error N=4 (5 Chebyshev polynomials) P 4 s(x) - s(x) (truncation error) I 4 s(x) - s(x) (interpolation error) I 4 s(x) - P 4 s(x) (aliasing error) y x ŝ n s n = , , , ,

30 Matrix of the differential operator 29 The matrices of derivative operators with respect to the Chebyshev basis (T 0, T 1, T 2, T 3, T 4 ) are d dx = d 2 dx 2 = so that the matrix of the differential operator d2 dx 2 4 d + 4 Id on the r.h.s. of Eq. (8) dx is A kl =

31 Resolution by means of a Galerkin method 30 Galerkin basis : φ 0 (x) := T 2 (x) T 0 (x) = 2x 2 2 φ 1 (x) := T 3 (x) T 1 (x) = 4x 3 4x φ 2 (x) := T 4 (x) T 0 (x) = 8x 4 8x 2 Each of the φ i satisfies the boundary conditions: φ i ( 1) = φ i (1) = 0. Note that the φ i s are not orthogonal Transformation matrix Chebyshev Galerkin: φki = such that φ i (x) = 4 φ ki T k (x). 4 2 Chebyshev coefficients and Galerkin coefficients: u(x) = ũ k T k (x) = ũ i φ i (x) The matrix φ ki relates the two sets of coefficients via the matrix product ũ = φ ũ i=0

32 Galerkin system 31 For Galerkin method, the test functions are equal to the trial functions, so that the condition of small residual writes (φ i, Lu s) = 0 3 (φ i, Lφ j ) ũ j = (φ i, s) j=0 with (φ i, Lφ j ) = = = ( φ ki T k, L φ lj T l ) = l=0 4 l=0 4 l=0 φ ki φlj (T k, 4 m=0 4 4 l=0 A ml T m ) = φ ki φlj π 2 (1 + δ 0k)A kl = π 2 φ ki φlj (T k, LT l ) l=0 φ ki φlj 4 m=0 A ml (T k, T m ) 4 (1 + δ 0k ) φ ki A kl φlj l=0

33 Resolution of the Galerkin system 32 In the above expression appears the transpose matrix [ Q ik := t (1 + δ 0k ) φ ] ki = The small residual condition amounts then to solve the following linear system in ũ = (ũ 0, ũ 1, ũ 2 ): Q A φ ũ = Q s with Q A φ = and Q s = The solution is found to be ũ 0 = , ũ 1 = , ũ 2 = The Chebyshev coefficients are obtained by taking the matrix product by φ : ũ 0 = , ũ 1 = , ũ 2 = , ũ 3 = , ũ 4 =

34 Exact solution Galerkin Comparison with the exact solution N = 4 33 y = u(x) x Exact solution: u(x) = e x sinh 1 sinh 2 e2x e 1 + e 2

35 Resolution by means of a tau method 34 Tau method : trial functions = test functions = Chebyshev polynomials T 0,..., T 4. Enforce the boundary conditions by additional equations. Since T n ( 1) = ( 1) n and T n (1) = 1, the boundary condition operator has the matrix b pk = ( The T n s being an orthogonal basis, the tau system is obtained by replacing the last two rows of the matrix A by (10): ũ 0 ũ 1 ũ 2 ũ 3 ũ 4 ) = The solution is found to be ũ 0 = , ũ 1 = , ũ 2 = , ũ 3 = , ũ 4 = ŝ 0 ŝ 1 ŝ (10)

36 Exact solution Galerkin Tau Comparison with the exact solution N = 4 35 y = u(x) x Exact solution: u(x) = e x sinh 1 sinh 2 e2x e 1 + e 2

37 Resolution by means of a pseudospectral method 36 Pseudospectral method : trial functions = Chebyshev polynomials T 0,..., T 4 and test functions = δ(x x n ). The pseudospectral system is 4 LT k (x n ) ũ k = s(x n ) 4 4 A lk T l (x n ) ũ k = s(x n ) l=0 From a matrix point of view: T A ũ = s, where / 2 0 1/ 2 1 T nl := T l (x n ) = / 2 0 1/

38 Pseudospectral system 37 To take into account the boundary conditions, replace the first row of the matrix T A by b 0k and the last row by b 1k, and end up with the system ũ 0 ũ 1 ũ 2 ũ 3 ũ 4 = 0 s(x 1 ) s(x 2 ) s(x 3 ) 0 The solution is found to be ũ 0 = , ũ 1 = , ũ 2 = , ũ 3 = , ũ 4 =

39 Comparison with the exact solution Exact solution Galerkin Tau Pseudo-spectral N = 4 38 y = u(x) x Exact solution: u(x) = e x sinh 1 sinh 2 e2x e 1 + e 2

40 Numerical solutions with N = Exact solution Galerkin Tau Pseudo-spectral y = u(x) x

41 Exponential decay of the error with N 40 0 log 10 u num -u exact oo Galerkin Tau Pseudospectral N = number of Chebyshev polynomials -1

42 Not discussed here Spectral methods for 3-D problems Time evolution Non-linearities Multi-domain spectral methods Weak formulation

43 42 4 Spectral methods in numerical relativity

44 Spectral methods developed in Meudon 43 Pioneered by Silvano Bonazzola & Jean-Alain Marck (1986). Spectral methods within spherical coordinates 1990 : 3-D wave equation 1993 : First 3-D computation of stellar collapse (Newtonian) 1994 : Accurate models of rotating stars in GR 1995 : Einstein-Maxwell solutions for magnetized stars 1996 : 3-D secular instability of rigidly rotating stars in GR

45 LORENE 44 Langage Objet pour la RElativite NumeriquE A library of C++ classes devoted to multi-domain spectral methods, with adaptive spherical coordinates : start of Lorene 1999 : Accurate models of rapidly rotating strange quark stars 1999 : Neutron star binaries on closed circular orbits (IWM approx. to GR) 2001 : Public domain (GPL), Web page: : Black hole binaries on closed circular orbits (IWM approx. to GR) 2002 : 3-D wave equation with non-reflecting boundary conditions 2002 : Maclaurin-Jacobi bifurcation point in GR

46 45 Code for producing the figures of the above illustrative example available from Lorene CVS server (directory Lorene/Codes/Spectral), see

47 Spectral methods developed in other groups 46 Cornell group: Black holes Bartnik: quasi-spherical slicing Carsten Gundlach: apparent horizon finder Jörg Frauendiener: conformal field equations Jena group: extremely precise models of rotating stars, cf. Marcus Ansorg s talk

48 Textbooks about spectral methods 47 D. Gottlieb & S.A. Orszag : Numerical analysis of spectral methods, Society for Industrial and Applied Mathematics, Philadelphia (1977) C. Canuto, M.Y. Hussaini, A. Quarteroni & T.A. Zang : Spectral methods in fluid dynamics, Springer-Verlag, Berlin (1988) B. Mercier : An introduction to the numerical analysis of spectral methods, Springer- Verlag, Berlin (1989) C. Bernardi & Y. Maday : Approximations spectrales de problmes aux limites elliptiques, Springer-Verlag, Paris (1992) B. Fornberg : A practical guide to pseudospectral methods, Cambridge University Press, Cambridge (1998) J.P. Boyd : Chebyshev and Fourier spectral methods, 2nd edition, Dover, Mineola (2001) [web page]