CLASSROOM NOTES PART II: SPECIAL TOPICS. APM526, Spring 2018 Last update: Apr 11

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1 CLASSROOM NOTES PART II: SPECIAL TOPICS APM526, Spring 2018 Last update: Apr 11 1

2 Function Space Methods General Setting: Projection into finite dimensional subspaces t u = F (u), u(t = 0) = u I, F : B 0 ( B 1 ) B 1, P N : Projection into an N dimensional subspace B N of B 1. Obtain a finite dimensional evolution equation (= an ODE system). P N : B 1 B N ( B 1 ), B N = span{φ 1,.., φ N } u u N B N, t u N = P N F (u N ), u N (t = 0) = P N u I 2

3 Readings: D. Gottlieb, S. Orszag: Numerical analysis of spectral methods, CBMS-NSF, Conference series, SIAM (1981). B. Fornberg: A practical guide to pseudo - spectral methods, Cambridge Monographs on Appl. Math., Cambridge University Press (1996). Canuto, Quarteroni, Hussaini, Zhang: Spectral methods and applications, Springer (1996). 3

4 The Galerkin projector: B 1 Hilbert space: Scalar product in B 1 u, v {φ n } orthonormal system (ONS): φ n, φ m = δ nm u P N u = min{ u v : v B N }, u := u, u Implies: u P N u B N {φ n } ONS P N u = Expansion into general ONS: t u = F (u), u N = N n=1 a n (t)φ n n φ m, φ n t a n =< φ m, F ( n N n=1 c n φ n, c n = φ n, u a n φ n ) >, m = 1 : N M mn = φ m, φ n : mass matrix, ONS: < φ m, φ n >= δ mn Example: The viscous Burger equation. 4

5 A sketch of convergence theory: Difference schemes: Stability + Consistency Convergence. SM: Stability + Approximation Convergence. t u = Lu, u(t = 0) = u I, t u N = P N L(u N ), u N (x, t = 0) = P N u I The global error: e = u N u The local error: z = P N u u (interpolated exact solution). Split: e = y + z, y = u N P N u Stability: t y = P L(u N u) = P Ly + P Lz, y(0) = 0 t y = P Ly + f, y, f B N y K f e K P Lz K P L P N u u Approximation: P N u u = α(n) e = O(α(N)) 5

6 The approximation error for Fourier series: u(x) = n Remainder: û(n)e inx û(n) = 1 2π û(n) = 1 1 2π ( in) p n >N e inx p xu(x) dx û(n)e inx = O(N 1 p ) p xu L 1 implies α(n) = O(N 1 p ) u(x)e inx dx 6

7 The interpolation projector - Collocation and Pseudo - spectral methods Problem: Cannot evaluate φ n, F (u N ) with Galerkin for general F. Example: t u = xu 2 e u, f, g = φ n, F (u) = π π f g dx, e inx m ( m 2 a m e imx ) dx φ n = e inx e inx exp[ m a m e imx ] dx Interpolation projector: Choose points x 1,.., x N ( collocation points, nodes, etc.) P N u = N n=1 a n φ n, P N u(x j ) = u(x j ), j = 1,.., N 7

8 (N ODEs in N unknowns for the a n ). t u N = P N F (u N ), t u N (x j ) = P N F (u N )(x j ) = F (u N )(x j ), j = 1,.., N N m=1 t a m e imx j = m ( m 2 a m e imx j) exp[ m a m e imx j], j = 1,.., N Change of basis and cardinal functions: Parametrize u N not by a n but by its values at x j. Example: t u = 2 xu e u, t u N (x j ) = 2 xu N (x j ) e u N(x j ), t u j = n u n 2 xψ n (x j ) e u j Re - write the interpolation operator as a Galerkin projector f, g = N n=1 w n f(x n )g(x n ) 8

9 Cardinal functions: ψ m (x) B N, ψ m (x j ) = δ mj, ψ m (x) = N n=1 φ n (x) ˆΨ(n, m) u N (x) = m u m ψ m (x) = n a n φ n (x) u N (x j ) = u j t u j = F (u N )(x j ) The differentiation matrix: Write collocation scheme as difference scheme with full stencil. t u j = n D 2 (j, n)u n e u j, D 2 (j, n) := 2 xψ n (x j ) 9

10 In general: 1. Choose basis functions φ n 2. Compute cardinal functions ψ n ψ n (x) = N m=1 φ m (x)ψ(m, n), ψ n (x j ) = δ nj 3. Compute differentiation matrices D 1 (j, n) = x ψ n (x j ), D 2 (j, n) = 2 xψ n (x j ), Replace derivatives k x by the matrices D k. 10

11 Choices: 1. How to choose finite dimensional subspace B N (the φ n ). 2. How to choose projection (Galerkin or interpolation). 3. How to choose collocation nodes x j, j = 1,.., N. 11

12 Choose x n as Gaussian integration nodes with a certain weight function in [a, b]: b a ω(x)p(x) dx = N n=1 w n p(x n ), p P 2N 1 Gaussian integration: Choose x n as the zeros of the N th orthogonal polynomial p N P N 1, p N (x n ) = 0, n = 1 : N 12

13 SUMMARY: Spectral methods and Galerkin: α(n) higher than polynomial in N. Example: Fourier series. Spectral methods and Collocation: Nodes x j have to be chosen such that interpolation converges for N. How to choose the collocation nodes:, weighted scalar product f, g = w(x)f (x)g(x) dx Gaussian integration rule: N j=0 ω j φ m (x j ) φ n (x j ) = δ mn, n, m = 0,.., N 13

14 Orthogonal polynomials: 1. Finite interval [a, b] = [ 1, 1]: (1a) ω = 1, Legendre polynomials (1b) ω = 1, Chebyshev polynomials 1 x 2 2. Infinite intervals (2a) [a, b] = R, ω = e x2 /2, Hermite polynomials (2b)[a, b] = [0, ), ω = e x, Laguerre polynomials 14

15 3. On a sphere: Spherical Harmonics x = r(y, 1 y 2 cos α, 1 y 2 sin α), y [ 1, 1], α [ π, π] S mn (x) = L mn (y)(1 y 2 ) n/2 e inα L mn : associated Legendre polynomials Remark: Spectral methods yield full matrices fast transform methods. 15

16 Fast Fourier Transform (FFT): Discrete Fourier Transform: u(n x) = ξ N 1 2π ν= N û(ν ξ) = x N 1 2π n= N û(ν ξ)e inν x ξ u(n x)e inν x ξ for N x ξ = π The differentiation matrix for a Fourier pseudo spectral method: (D k u)(n x) = ξ N 1 2π ν= N (iν ξ) k û(ν ξ)e inν x ξ 16

17 gives in matrix form D k = ΩΛΩ 1 Ω(n, ν) = ξ 2π e inν x ξ, Λ = [diag(iν ξ) k ] direct evaluation of D k : the product Du needs O(N 2 operations. the products Ωu, Ω 1 û can be done in O(N log 2 N) operations, using FFT! Λ diagonal N operations. 17

18 The FFT concept: How to compute the terms y(n) = renumber x: M = JK M 1 m=0 x(m) exp[inm 2π M ] x(j + Jk) = z(j, k), j = 0 : J 1, k = 0 : K 1 y n = J 1 j=0 exp[inj 2π K 1 JK ] k=0 z(j, k) exp[ink 2π K ] 18

19 compute: v(n, j) = K 1 k=0 z(j, k) exp[ink 2π ], n = 0 : K 1, j = 0 : J 1 K Periodicity: v(n + K, j) = v(n, j)!!! and y(n) = J 1 j=0 exp[inj 2π ]v(n, j), n = 0 : JK 1 JK total: M(J + K) operations instead of MJK operations! Recursion: set K = LS and z(j, k) = z 1 (j, l, s) v(n, j) = K 1 k=0 z(j, k) exp[ink 2π ], n = 0 : K 1, j = 0 : J 1 K perform this step in JK(L + S) instead of JK 2 operations! 19

20 RIEMANN SOLVERS (GENERAL IDEA) t u(x, t) + x f(u, x, t) = 0 Assume u(x, t) piecewise constant. cell n. u(x, t) = u n (t)forx Solve t u(x, t) + x f(u, x, t) = 0 exactly, computing ũ(x, t + t). Compute u(x, t + t), piecewise constant on the cells, by averaging ũ(x, t + t). 20

21 THE METHOD OF CHARACTERISTICS (semilinear) Consider the hyperbolic equation t u + a(x) x u = b(x, u), Consider the ODE system d dtξ(y, t) = a(ξ), ξ(y, 0) = y d dt w(y, t) = b(ξ, w), w(y, 0) = ui (y) this implies u(ξ(y, t), t) = w(y, t) for all y u(x, 0) = u I (x) To compute the solution at (x, t), this process has to be inverted. Find the general solution of d dt ξ(y, t) = a(ξ), ξ(y, 0) = y d dt w(y, t) = b(ξ, w), w(y, 0) = ui (y) For a given x, t find y such that ξ(y, t) = x. Set u(x, t) = w(y, t) 21

22 THE METHOD OF CHARACTERISTICS (nonlinear) Consider the hyperbolic equation t u + a(x, u) x u = b(x, u), Consider the ODE system d dtξ(y, t) = a(ξ, w), ξ(y, 0) = y d dt w(y, t) = b(ξ, w), w(y, 0) = ui (y) this implies u(ξ(y, t), t) = w(y, t) for all y u(x, 0) = u I (x) To compute the solution at (x, t), this process has to be inverted. Find the general solution of d dt ξ(y, t) = a(ξ, w), ξ(y, 0) = y d dt w(y, t) = b(ξ, w), w(y, 0) = ui (y) For a given x, t find y such that ξ(y, t) = x. Set u(x, t) = w(y, t) 22

23 d dt ξ(y, t) = a(ξ, w), ξ(y, 0) = y d dt w(y, t) = b(ξ, w), w(y, 0) = ui (y) Example: Burger s equation (nonlinear autonomous) t u + x ( 1 2 u2 ) = 0, a(x, u) = u, b(x, u) = 0 yields multiple solutions (dependent on the form of u I (x))! In the nonlinear case characteristics can intersect Discontinuities (shocks) develop. 23

24 THE CONCEPT OF WEAK SOLUTIONS t u + x f(u) = 0, u(x, 0) = u I (x) (1) u discontinuous need to re- define derivatives ψ sufficiently smooth test function, ψ C 0 Definition: u is a weak solution of (1) iff 0 dt dx [u t ψ + f(u) x ψ] = dx [ψ(x, 0)u I (x)] ψ Remark: u differentiable u classical solution. Example: t u + x [H(x)u + 1 H( x)u] = 0, 2 u(x, 0) = ui (x) 24

25 THE RANKINE - HUGONIOT CONDITION THE HUGONIOT LOCUS Relates shockheight to shockspeed if there is a shock. t u + x f(u) = 0, Assume: 1. u differentiable away from the shock curve x = γ(t). 2. u is a weak solution. Implies: γ (t)(u + u )(γ(t), t) = (f + f )(γ(t), t) Riemann problem: t u + x f(u) = 0, ( u for x < 0, u(x, 0) = u + for x > 0, ), u > u + 25

26 RAREFACTION WAVES AND THE BARENBLATT SOLU- TION Riemann problem: t u + x f(u) = 0, ( ) u for x < 0, u(x, 0) =, u u + for x > 0, < u + Characteristics do not cover the x, t domain! t u + x f(u) = 0, set u(x, t) = g( x t ) f (g(z)) = z u(x, t) = (f ) 1 ( x t ) gives continuous solution! For this to work f has to be convex! Example (Burger): f(u) = 1 2 u2 26

27 GODUNOV S METHOD Scalar problems: Equation for averages: t u + x f(u) = 0 U n j = 1 x xj+1 x j u(x, t n )dx, F n j = 1 t tn+1 t n f(u(x j, t))dt. U n+1 j U n j + t x (F n j+1 F n j ) = 0 Numerical approximation: compute F n j from U n j 1, U n j Piecewise constant approximation: Assume u(x, t n ) = Uj n for x j < x < x j+1. Solve problem with piecewise constant initial data (the Riemann problem) exactly. 27

28 GODUNOV AND THE RIEMANN PROBLEM t u + x f(u) = 0, u(x, 0) = ( UL for ) x < 0 U R for x > 0 For Godunov s method we have to compute F = t 1 from the cell averages U L, U R. t 0 f(u(0, t))dt Characteristics: x = f (u), u = 0 Characteristic slopes given by f (U L ), f (U R ) Five cases: 1-4: Shock curve according to Rankine - Hugoniot: γ = f(u R) f(u L ) U R U L 5: rarefaction wave 28

29 Case 1: f (U R ) > 0, f (U L ) > 0 F = f(u L ) Case 2: f (U R ) < 0, f (U L ) < 0 F = f(u R ) Case 3: f (U R ) < 0 < f (U L ) and γ > 0 F = f(u L ) Case 4: f (U R ) < 0 < f (U L ) and γ < 0 F = f(u R ) Case 5: f (U L ) < 0 < f (U R ): rarefaction wave and the Barenblatt solution. 29

30 GODUNOV S METHOD: γ := f(u n j ) f(u n j 1 ) U n j U n j 1 Case 1: f (U n j ) > 0, f (U n j 1 ) > 0 F n j = f(u n j 1 ) Case 2: f (U n j ) < 0, f (U n j 1 ) < 0 F n j = f(u n j ) Case 3: f (U n j ) < 0 < f (U n j 1 ) and γ > 0 F n j = f(u n j 1 ) Case 4: f (U n j ) < 0 < f (U n j 1 ) and γ < 0 F n j = f(u n j ) Case 5: f (U n j 1 ) < 0 < f (U n j ) F n j = f(u s) with f (u s ) = 0. u s : sonic point 30

31 STABILITY IN W 1 1 THE NORM (THE TVD PROPERTY) u W q p = q xu Lp = ( q xu p dx) 1 p Non-oscillatory schemes: Definition: A scheme is TVD j (T 1)Uj n+1 j (T 1)U n j The linear case: Write scheme solely in terms of derivatives. U n+1 = U n T 1 (A n V n ) + B n V n, Theorem: A, B 0, A + B 1 T V D V n := (T 1)U n Example: Lax-Friedrichs TVD, Lax Wendroff not TVD Problem: TVD methods are maximally second order! 31

32 FLUX LIMITER METHODS U n+1 = U n c(t 1)F n Idea: Lower order method with numerical flux F L. Higher order method with numerical flux F H. Lower order method nonoscillatory. Combine: F = F L + Φ(F H F L ) Smooth part: φ 1; Non-oscillatory otherwise, i.e. total method is TVD. Total method formally only first order but higher order in smooth regions. 32

33 THE FLUX LIMITER Condition 1: (Smooth order) Φ = Φ( (1 T 1 )U (T 1)U ), φ(1) = 1 Condition 2: Choose φ such that total method is TVD. Derive φ for linear case and use in general. 33

34 t u + a x u = 0, a > 0 Example: Upwind and Lax Wendroff U n+1 = U n (T 1)F n upwind (for a > 0): U n+1 = U n a t x (1 T 1 )U n upwind flux F L = νt 1 U = νu νt 1 V with ν = a t x = ac and 0 ν 1 because of the CFL condition. Lax-Wendroff: U n+1 = U n 2 x a t (T T 1 )U n + a2 t 2 2 x 2 (T 2 + T 1 )U n Lax -Wendroff flux F H = ν 2 [(1+T 1 )U ν(1 T 1 )U] = νt 1 U + ν 2 (1 ν)t 1 V 34

35 Flux limiter flux F = F L +(F H F L )φ = νu νt 1 V + φ ν 2 (1 ν)t 1 V, V = (T 1)U (T 1)F = (T φ) ν 2 (1 ν)v + νt 1 V φ ν 2 (1 ν)t 1 V TVD: U n+1 = [U T 1 (AV ) + BV ] n and φ = φ(θ), θ = V T 1 V T φ ν 2 (1 ν)θt 1 V +νt 1 V φ ν 2 (1 ν)t 1 V = T 1 (AV ) BV set B = 0, T 1 A = (T φ) ν 2 (1 ν)θ + ν φν (1 ν), 0 A ν [φ(t θ)(1 ν)θ + 1 φ(θ)(1 ν)] 1, ν [0, 1], θ 2 0 ψ(ν, θ 1, θ 2 ) 2 for all ν [0, 1] and all θ 1, θ 2 ψ(ν, θ 1, θ 2 ) = ν[φ(θ 2 )(1 ν)θ φ(θ 1 )(1 ν)] 35

36 Two standard choices: 1. Superbee: φ(θ) = max{0, min{1, 2θ}, min{θ, 2}} 2. Van Leer: φ(θ) = θ + θ 1 + θ 36

37 SPECIAL TOPICS: t u + x f(u) = 0 LEAPFROG AND THE YI CELL: u n+1 = u n 1 t 2 x (T T 1 )f(u n ) von Neumann stability: plane waves for f(u) = au u(x, t) = exp[iξ(x vt) + rt] T T 1 = e iξ x e iξ x = 2i sin(ξ x), λ 1 λ û n+1 û n 1 = 2iαû n = 2iα, α = a t x λ 2 + 2iα 1 = 0, λ 12 = iα ± sin(ξ x) a tξ 1 α 2, λ 1 iα + (1 α2 2 ), λ 2 iα (1 α2 2 ) 37

38 λ 1 iα + (1 α2 2 ), λ 2 iα (1 α2 2 ) λ 1 real mode, λ 2 spurious mode has to be kept small by the initial conditions! λ 12 2 = ( iα + 1 α 2 )(iα + 1 α 2 ) = 1 for α 1 λ 12 = e r 12 t = 1 r 12 = 0 no damping and no artificial diffusion! 38

39 LEAPFROG FOR THE SECOND ORDER WAVE EQUA- TION t u = a x w, t w = b x u 2 t u = ab 2 xu u n+1 u n = a t x (T 1 2 T 1 2)w n, w n+1 w n = b t x (T 1 2 T 1 2)u n+1, u n+1 2u n + u n 1 = ab t2 x 2(T 1 2 T 1 2) 2 u n plane wave: u(x, t) = e iξ(x vt)+rt, λ = e rt λ 2 2λ+1 = abα 2 λ, α = t x [eiξ x/2 e iξ x/2 ] = 2i t x sin(ξ x/2) λ 2 2λ+1 = 4βλ, 4β = abα 2 = 4ab t2 x 2 sin2 (ξ x/2), 0 β 1 λ 2 2(1 2β)λ+1 = 0, λ 12 = (1 2β)±i Again, no artificial diffusion! 4β(1 β) λ 12 = 1, r = 0 39

40 the Yi cell: t u = A x w, t w = B x u, u, w, x R 3 u n+1 = u n + t x A(T 1 2 T 1 2)w n, w n+1 = w n + t x B(T 1 2 T 1 2)u n+1 Application: Maxwell s equations A x w = curl(w), B x u = curl(u) t u = curl(w), t w = curl(u), u, w, x R 3 u : electric field, w : magnetic field 40

41 BOUNDARY CONDITIONS AND GHOSTPOINTS FOR HYPERBOLIC SYSTEMS t u + x [A(x, t)u] = 0, x 0, Bu(0, t) = u b, u R N, B R K N Necessary condition for well posedness: Relation of B to A: Influx given in terms of outflux! Diagonalization: A(0, t) = EDE 1, D = Important!! dim(d + ) = K K Projections: partition E ( D+ 0 0 D E = (E +, E ), E + : N K, E 1 = ( R+ R ), D + > 0, D 0 ), R + : K N BE + : K K, Necessary: (BE + ) 1 41

42 Ghostpoint: Choose extra gridpoint(s) x. Diagonalize locally and split and use extrapolation. Example: three point stencil, second order method. V = ( V+ V ( ) V+ V U( x) = 2E ) = ( R+ R ( x) = ) U, U = ( E +, E ) ( V + V ( ) 2V b + V + ( x) 2V (0) V ( x) V b + = (BE +) 1 [u b BE V (0)] ( (BE+ ) 1 [u b BE R U(0)] R U(0) ) ) U( x) Use U( x) to compute U(0, t + t) with the difference method. Important: the accuracy of the extrapolation has to be at least as high as the accuracy of the difference method. 42

43 BOUNDARY CONDITIONS AND LAGRANGE MULTIPLI- ERS Concept for constrained ODEs: Mȧ = G(a), a R N constraint (BC) : B T a = b(t), B R N K, b R K (M: mass matrix), define F (a) = M 1 G(a) ȧ = F (a) ȧ = P M 1 G(a) P : projection of ȧ on the manifold M = {u : B T u = ḃ} y T (P u u) = 0, y : B T y = 0 and B T P u = ḃ P u u = Bλ, λ R K and B T P u = ḃ P u = [I B(B T B) 1 B]u + B(B T B) 1 ḃ projected equation: ȧ = [I B(B T B) 1 B T ]F (a) + B(B T B) 1 ḃ B T ȧ = ḃ B T a(t) = b(t) for all time if B T a(0) = b(0) 43

44 Spectral methods: M mn = φ m, φ n, G m = φ m, F ( n a n φ n ), Galerkin: f, g = ωfg dx, Collocation: f, g = j w j f(x j )g(x j ) Example: (Dirichlet BC for x [α, β]) B(n, 1) = φ n (α), B(n, 2) = φ n (β) A special case: collocation points include the end points (x 1 = α, x N = β) reduces to ȧ 1 = t u(x 1 ) = ḃ α, ȧ N = t u(x N ) = ḃ β Lobatto points: Force x 1 = 1, x N = 1 by reducing the degree of the polynomials for which Gauss integration is exact by 2. 44