256 Summary. D n f(x j ) = f j+n f j n 2n x. j n=1. α m n = 2( 1) n (m!) 2 (m n)!(m + n)!. PPW = 2π k x 2 N + 1. i=0?d i,j. N/2} N + 1-dim.

Transcription

1 56 Summary High order FD Finite-order finite differences: Points per Wavelength: Number of passes: D n f(x j ) = f j+n f j n n x df xj = m α m dx n D n f j j n= α m n = ( ) n (m!) (m n)!(m + n)!. PPW = π k x ν = k c t π Phase error: Leading term of the relative error. Often ( ) order π PE(p, ν) Cν. PPW Work per wavelength: W m = m PPW t t, where m = order. Infinite-order finite differences: As above with m. Demand exactness for trig. polynomial e ilx. Find coefficients by comparing with Fourier series for x x. Rearranging the sum gives du dx x j = N [ ( )] π ( )j+i sin (j i) u i. N + i=0?d i,j Trigonometric Polynomial Approximation Assume u: [0, π] R periodic. N even. Spaces: BˆN 4 span{einx { ( : n )} N/} N + -dim. N B N 4 Bˆn \ sin x N-dim.. Continuous Expansion Fourier series: P N u(x) = û n = π n= π 0 û n e inx, f(x)e inx dx.

2 Section Special cases: u real û n = û n, u even only cosines, u odd only sines. Approximation: n= û n < u P N u L 0. n= û n < u P N u L 0. u (0 m ) (viewed periodically) is continuous, u (m) L û n (/n) m. Spectral convergence: u C û n decays faster than any power of n. PD = DP. Projection and differentiation commute. (start with expansion above, carry out both.) Truncation error: P N L(Id P N )=0... Approximation Theory for the Continuous Expansion Sobolev norm: Parseval s Identity: h = /N. u H r : q u q = m=0 n D m u L Proof: Parseval, consider tail, smuggle in an n q u analytic: n= û n = π u. π 0 u P N u L Ch q u (q). L n q. u P N u L Ce cn u L Proof: u (q) L Cq! u,stirling s Formula: q! q q e q, q N. L u H r : u P N u H q Ch r q u H r. û n (+ n ) q. Proof: Parseval, (+ n ) q (+ n r ). (r q) N u C q, q > /: u P N u L h q / u (q). L Proof: u P N u, smuggle in n q, CSU. L a constant coefficient differential operator:. Discrete Expansion.. Discrete Even Expansion x j =πj/n, j = 0 N. (N points) s Lu = j= a j d j u dx j. Lu LP N u H q h r q s u H r.

3 Trigonometric Polynomial Approximation 3 Exactness: Periodic case: Trapezoidal rule is Gauß quadrature. Proof: Evaluate geometric series. Coefficients: u BˆN : ũ n = Nc n π u(x)= π 0 N N u(x j ) N e inxj u(x j ), where c n =+ n=n/ to compensate for ũ N/ =ũ N/. N coefficients, N quadrature points. Interpolant: with I N : L B N. I N u(x) = g j (x) = N sin ( n N/ N = ũ n e inx. g j (x)u(x j ) N x x ) j cot I N u(x j )=u(x j ). (rewrite sums, geometric series) ( x xj Two different ways to differentiate: go through mode space or don t. Differentiation matrix is circulant. sin N/ consequences: I N d dx DI N (d/dx: B N B N) D D (). Spatial discretization does not cause phase error deterioration... Discrete Odd Expansion x j =πj/(n +) j = 0 N. (N + points) Exactness: Periodic case: Trapezoidal rule is Gauß quadrature. Coefficients: Interpolant: u BˆN: π u(x) = π 0 N + ũ n = N + J N u(x) = ). N u(x j ). N u(x j )e inxj. n N/ N = l=0 ũ n e inx u(x l )h l (x) with J N : L BˆN. h l (x)= sin N + sin ( N + ( ) (x x l ) N/ ) = (x x l) k= N/ e ik(x xl).

4 4 Section 3 J N u(x j ) =u(x j ). May also be viewed as Lagrange trigonometric interpolant: Same differentiation matrix as -order FD. I N d dx = DI N...3 Approximation Theory for Discrete Expansions u H q, q >/: c nũ n =û n + m,m 0 û n+nm Proof: Substitute continuous into discrete, exchange sums because of absolute convergence, smuggle+csu. Aliasing error: u H r, r >/: Proof: smuggle, CSU. u H r, r >/: A N u 4 c nũ n û n. A N u L h r u (r). L u I N u L h r u (r). L Proof: Error = aliasing+truncation. u H r, r >/: u H r, r >/: A N u H q h r q u H r. u I N u H q h r q u H r, Lu LI N u H q h r q s u H r. 3 Fourier Spectral Methods Consider u t = Lu. 3. Fourier Galerkin Defining assumption: R N = t u N Lu N BˆN. Build method: Calculate residual, project onto BˆN, set to zero. Multiplication (for nonlinear problems) becomes convolution. (e.g. Burgers) More complicated nonlinearities: no way. Very efficient for linear, constant-coefficient problems with periodic BCs. 3.. Stability L semi-bounded: stability. L+L αid

5 Fourier Spectral Methods 5 Proving semi-boundedness: Integrate by parts. Examples: L=a(x) x L= x b(x) x L semi-bounded Fourier-Galerkin stable. Proof: show P N = P N by (P N u, v)=(p N u, P N v). Then L N = P N LP N semi-bounded. 3. Fourier Collocation Defining assumption: R N yj = 0 Optionally: Collocation points {y j } Quadrature points {x j }. (we won t do that) Build method: Expand u with Lagrange interpolation polynomial. Obtain residual. Set to zero at collocation points simply replace derivatives by application of the differentiation matrix. 3.. Stability I N I N, so Fourier Galerkin proof breaks. Discrete inner product: u N N = u N L for odd expansion. u N N u N L for even expansion. L = a(x)u(x), 0 </k a(x) k: (u, v) N = N + N f(x j )g(x j ) u N (t) N k u N (0). Proof: Multiply by u N /a, obtain (/a)d/dt( u ). Use exactness of quad. formula, periodicity to get d/dt =0. Exploit boundedness of a. u = A D u: Use A / as a change of variables, then bound u = e ADt u 0 by saying A / D A / is skew-symmetric. Proof remains valid for u = DAu, L= a(x), L = a(x)u(x) with a(x) changing sign, but a x / α uniformly treat skew-symmetric form Lu= a u x+ (a u) x a xu to get u N N e αt u 0 N : Proof: Multiply by u N, get d/dt u N. Integrate (exact) by parts in the second term, only third term left over, yields bound. skew-symmetric equation can be written u N t + J Na x u N + xj N [a u N ] J N(a x u N ) = 0, u N t + J Na x u N + xj N [a u N ] (J N x (a u N ) J N a x u N ) = 0, u N t + J Na x u N + xj N [a u N ] J N x (au N )?A N 4 = 0 A N L h s un (s) L

6 6 Section 4 (it s s because A N contains derivatives). This motivates the......superviscosity method L u = Lu +( ) s ε N s x s u N. Stable if ε > some constant C. (s) Proof: Add A N on both sides, integrate (u N, A N ) N by parts, u N L. Bound superviscosity term by same norm, bound for (u, t u) N involving a x shows up. Using Fourier Galerkin, see that superviscosity = filtering. L = b(x) x u, b > 0: matrix method: Define D () = D, note D u BˆN, D () real u B N, use skew-hermiticity. integral method: x 4 I N x I N x I N, then rewrite as integral. L = f(u) x : Spectral viscosity method where Q m is a filter Superspectral viscosity method t u N + x P N f(u N )=ε N ( ) s+ x s [Q m x s u N ] t u N + x P N f(u N ) =ε N ( ) s+ x s u N. 4 Orthogonal Polynomials B N 4 span{x n : 0 n N}. Fourier methods achieve exponential accuracy only if u is periodic. Sturm-Liouville operator: p > 0, 0 q <M, w the weight function. Parseval identity: Lϕ = x (p x ϕ)+ qϕ = λwϕ (u, u) Lw = γ n û n, γ n =(ϕ n, ϕ n ), û n = γ n (u, ϕ n ) Lw. Estimate decay of û n by plugging in eigenvalue problem, using selfadjointness of operator. Singular Sturm-Liouville problem: p vanishes at boundary. û n C ( ) m L u w λ n m spectral decay for C functions with zero BCs. (Regular problem: only for periodic problems, otherwise boundary causes error.) Jacobi polynomials: P n (α,β), α, β > p(x)=( x) α+ ( +x) β+, w(x) =( x) α (+x) β, q(x) =cw. Lw. Rodrigues formula: Derivative: ( x) α ( +x) β P n α,β (x) = n n! x n ( x) α+n ( + x) β+n. d dx P n (α,β) = n + α + β + P (α+,β+) n (x).

7 Polynomial Expansions 7 Odd/Even: P n (α,β) = ( ) n P n (α,β) ( x). There are various three-term recurrence for these polynomials, P (α,β) 0 =, P (α,β) = (α + β + )x + (α β)/. Legendre polynomials: α = β =0, w, called P n Chebyshev polynomials: p = x, q =0, w = p. T n = cos(n arccos(x)). x T n = T n + T n+. Chebyshev is best approximation to x n+ among polynomials of degree n. Ultraspherical/Gegenbauer polynomials: α = β. PPW for polynomials: 4. (Gegenbauer expansion, decay of the Bessel function) 5 Polynomial Expansions Can somewhat easily differentiate and integrate, requires three-term stuff and its inverse. Gauß-Lobatto quadrature: both endpoints part of the quadrature. Exact for B N. Gauß-Radau quadrature: one endpoint part of the quadrature. Exact for B N. Pure Gauß quadrature: no endpoints part of the quadrature. Exact for B N+. Each different kind of polynomial has a different set of quadrature points and weights because each has a different weight function. Chebyshev Quadrature: with GL ( ) GR ( ) G ( ) j j j + x j = cos N π w j = cos N + π z j = cos N + π w j = π v j = π u j = π c j N c j N + N + c j = + N + 0. j = 0,, N [, ] w denotes discrete inner product, N,w discrete norm. Discrete Gauß-Lobatto norm: not exact for n = N, but equivalent. Discrete Expansion: N I N u(x)= n=0 P n (α) (x)ũ n, ũ n = N u(x j )P (α) n (x j )w j γ n Quadrature points are interpolation points. Proof: Plug coefficient terms into expansion, exchange sums to find l j (x)=w j is the Lagrange interpolation polynomial. N n=0 P (α) n (x)p (α) n (x j ) γ n Differentiation matrices are nilpotent. (Decrease in order) GL Differentiation matrix is centro-antisymmetric. D (q) = D q.

8 8 Section 6 Runge phenomenon: Wild behavior of polynomials near interval boundaries. u C 0 [, ], {x j } interpolation nodes. Then u I n u +Λ N u p, where p is the best-approximating polynomial and Λ N C log(n + )+C. Cauchy interpolation remainder: Λ n = max λ n, λ n = N l j (x). [,] u(x) I N u(x)= u(n+) (ξ) (N + )! Grid points should cluster quadratically near the boundary. n (x x j ). 6 Polynomial Spectral Methods & Stability 6. Galerkin Defining assumption: Residual orthogonal to B N. Stiffness matrix: Mass matrix: positive definite because L -norm is a norm. Formulation: S k,n = ϕ k Lϕ n wdx. γ k M k,n = ϕ k ϕ n wdx, γ k ȧ =M Sa. Basis constructed as a linear combination of P n (α) to ensure BCs are kept. u t = Lu. If L is semi-bounded (L+L γid), then the Galerkin method is stable. Linear hyperbolic equation well-posed in Jacobi norm for α 0, β 0, but not for Chebyshev. (Consider x /ε. Norm blows up, because Cheb weights blow up.) 6. Tau Defining assumption: Residual orthogonal to B N k, where k is the number of BCs, demand that it is zero. BC coefficients can be obtained once PDE-discretizing coefficients are computed. Mass matrix remains diagonal. Usable for elliptic problems, allows efficient preconditioners. Burgers: Product once again becomes convolution-like term. 6.3 Collocation Defining assumption: Residual zero at interpolation/quadrature nodes.

9 Polynomial Spectral Methods & Stability 9 Stability: Usual go-to-integral stuff. 6.4 Penalty Method for Boundary Conditions Example: Q (x) = ( x)p N (x) P N ( ) { x=, = 0 x=x j. u N t + a u N x = τa Q (x)(u N ( ) BC) Consistent because exact solution satisfies scheme exactly. Stable: go back to integral, gives boundary values, tweak τ to be bigger than corresponding weight.