Third part: finite difference schemes and numerical dispersion

Transcription

1 Tird part: finite difference scemes and numerical dispersion BCAM - Basque Center for Applied Matematics Bilbao, Basque Country, Spain BCAM and UPV/EHU courses : Advanced aspects in applied matematics Topics on numerics for wave propagation (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

2 Linear transport equation Te simplest model for te wave propagation is te linear transport equation: u t + u x = 0, x R, t > 0, u(x, 0) = f (x). (1) u = u(x, t) is a solution of (1) iff u is constant along te caracteristic lines x + t = constant. Te solution of (1) is u(x, t) = f (x t). Semigroup teory for te transport equation (1). Te Hilbert space H := L 2 (R), te operator A := x and its domain D(A) := H 1 (R). A is dissipative. < Au, u > L 2 (R) = R x uu dx = 1 2 R x (u2 ) dx = 0. A is maximal. For any f L 2 (R), tere exists an unique solution u H 1 (R) of te equation u + x u = f, wic can be explicitly computed as x 0 u(x) = f (s) exp(s x) ds = f (z + x) exp(z) dz. By te Minkowski inequality u L 2 (R): u x = f u L 2 (R) u H 1 (R). Transport equation wit reversed sign, Te solution of (2) is u(x, t) = g(x + t). 0 u L 2 (R) f L 2 (R) exp(z) dz = f L 2 (R). u t u x = 0, x R, t > 0, u(x, 0) = g(x). (2) (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

3 Tree semi-discrete finite difference approximations of u t + u x = 0 forward u j (t) + u j+1 (t) u j (t) = 0, centered u j (t) + u j+1 (t) u j 1 (t) 2 = 0, backward u j (t) + u j (t) u j 1 (t) = 0. (3) Briefly, forward/centered/backward: u (t) = A u (t) / 1/ 0 A = / 1/... A =... 1/2 0 1/ /2 0 1/ / 1/ 0 0 A = / 1/ 0... (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

4 Peter Lax convergence results Peter Lax s equivalence teorem: CONVERGENCE CONSISTENCY+STABILITY. CONSISTENCY=insert a smoot solution of te continuous model in te discrete one + Taylor expansions. STABILITY = von Neumann analysis. Semi-discrete Fourier transform at scale : û (ξ, t) = j Z u j (t) exp( ix j ξ), ξ [ π/, π/]. All te tree scemes can be transformed into te first-order differential equation wose solution is û t (ξ, t) = p (ξ)û (ξ, t), û (ξ, 0) = û,0 (ξ) Here, p (ξ) = û (ξ, t) = û,0 (ξ) exp( p (ξ)t). 1 exp(iξ), forward sceme i sin(ξ), centered sceme exp( iξ) 1, backward sceme. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

5 Properties of te semi-discrete Fourier transform (SDFT) Definition Consider a sequence f := (f j ) j Z l 2 related to a grid of size (i.e. f j = f (x j )), its semi-discrete Fourier transform at scale is f (ξ) := f j exp( iξx j ), wit ξ [ π/, π/]. j Z Inverse Fourier transform: f j = 1 π/ f 2π (ξ) exp(iξx j ) dξ. π/ Remark Continuous Fourier transform: function f (x), x R, transformed into function f (ξ), ξ R. Semi-discrete Fourier transform: sequence f := (f j ) j Z transformed into function f (ξ), ξ [ π/, π/]. Parseval identity: f 2 l 2 := f j 2 = 1 2π f 2 L j Z 2 ( π/,π/) := 1 2π π/ f (ξ) 2 dξ. π/ Sannon sinc function: ψ 0 (x) = sin(πx/) πx/, wic is globally analytic and ψ 0 (0) = 1. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

6 Properties of te semi-discrete Fourier transform (SDFT) Te sinc function is a particular case of function f in te Paley-Wiener teorem: Teorem (Paley-Wiener, see Rudin, Real and complex analysis ) If A, C > 0 and f L 2 (C) is an entire function s.t. f (z) C exp(a z ) (exponential growt at most A), for all z C, ten te Fourier transform f of f as compact support in [ A, A]. Exercise: ψ 0 L 2 (R) and ψ 0(ξ) = χ ( π/,π/) (ξ), wit χ S te caracteristic function of set S. Definition Set ψ j (x) = ψ 0(x x j ) and te sequence f := (f j ) j Z l 2. Te continuous function f (x) := f j ψ j (x) is j Z called te sinc interpolation of f. Important properties of te sinc interpolation (exercise): f (ξ) = f (ξ) and f L 2 (R) = f l 2. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

7 Oter possible interpolations piecewise constant, using functions ψj 0(x) := χ [x j 1/2,x j+1/2 ]: f 0 (x) = f j ψj 0(x). j Z piecewise linear and continuous, using functions ψj 1(x) := (ψ0 j ψj 0)(x): f 1 (x) = f j ψj 1(x). j Z spline interpolation, using functions ψj m (x) = (ψj 0 ψj 0 )(x) (m successive convolutions): f m (x) = f j ψj m (x). j Z f 0 L 2 (R) = f l 2 and 1 3 f l 2 f 1 L 2 (R) f l 2. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

8 Observation ψ0 0 and ψ1 0 ave compact support, but teir Fourier transforms ψ 0 0 (ξ) = sinc(ξ/2) and ψ 0 1(ξ) = ( ψ 0 0(ξ))2 are spread. ψ 0 is spread, but its Fourier transform ψ 0 (ξ) = χ ( π/,π/) (ξ) as compact support. Cf. Heisenberg Uncertainty Principle, is not possible bot f and its Fourier transform f to ave compact support: Teorem (Heisenberg Uncertainty Principle, Stein & Sakarci, Fourier analysis - an introduction) Let f S(R), f L 2 (R) = 1. Ten ( )( x 2 f (x) 2 dx R R ) ξ 2 f (ξ) 2 dξ 1 16π 2, wit equality iff f (x) = A exp( B x 2 ), B > 0 and A 2 = 2B/π. In fact, for all x 0, ξ 0 R, ( )( ) x x 0 2 f (x) 2 dx ξ ξ 0 2 f (ξ) 2 dξ 1 16π 2. R R Interpretation in quantum mecanics. Te more certain we are about te location of a particle, te less certain we can be about its momentum and vice versa. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

9 Back to stabilization of numerical scemes... Set p = max ξ [ π/,π/] Re( p (ξ)). Using Parseval identity for te SDFT, u (t) 2 l 2 = 1 π/ û,0 (ξ) 2 exp(2tre( p (ξ))) dξ 2π π/. exp(2t p ) 1 π/ û,0 (ξ) 2 dξ = exp(2t p 2π ) u,0 2 l 2. π/ Numerical scemes (3) are stable iff p 0. Oterwise p 1/ and exp(2t p ) as 0. FORWARD: Re( p (ξ)) = 2 sin 2 (ξ/2)/ 0, ξ [ π/, π/], and p = 2/ UNSTABLE!!!. CENTERED: Re( p (ξ)) = 0, ξ [ π/, π/], and p = 0 STABLE. BACKWARD: Re( p (ξ)) = 2 sin 2 (ξ/2)/ 0, ξ [ π/, π/], and p = 0 STABLE. NECESSARY GEOMETRIC CONDITION FOR THE CONVERGENCE OF A NUMERICAL SCHEME: Te domain of dependence of te numerical sceme MUST CONTAIN te domain of dependence of te continuous model. Domains of dependence at a point (x j, t): Continuous transport: te segment joining (x j, t) wit (x j t, 0). Forward sceme: te semi-strip TO THE RIGHT of x = x j delimited by te times 0 and t. Centered sceme: te band delimited by te times 0 and t. Backward sceme: te semi-strip to te left of x = x j delimited by te times 0 and t. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

10 Error estimates Linear transport equation Consistency errors: Consider u a smoot solution of te transport equation u t(x, t) + u x (x, t) = 0, u(x, 0) = f (x), x R. Ten, by plugging u in te numerical sceme and using Taylor expansions, we obtain: backward sceme: u t(x j, t) + u(x j, t) u(x j 1, t) = u t(x j, t) + u x (x j, t) }{{} =0, u solves u t +u x =0 2 uxx (x j 1/2, t) := O j (t), x j 1/2 (x j 1, x centered sceme: u t(x j, t) + u(x j+1, t) u(x j 1, t) 2 = u t(x j, t) + u x (x j, t) }{{} =0, u solves u t +u x = (uxxx (x j 1/2, t) + uxxx (x j+1/2, t)), x j±1/2 (x j 1/2±1/2, x j+1/2±1/2 ). Set te error ɛ j (t) := u j (t) u(x j, t), were u j (t) is te solution of te backward sceme wit data u j (0) = f (x j ). Ten ɛ j (t) solves te problem ɛ j (t) + ɛ j (t) ɛ j 1 (t) = O j (t), ɛ j (0) = 0. (4) (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

11 Error estimates Linear transport equation ENERGY METHOD. Multiply (4) by ɛ j (t) and add in j Z: 1 d [ ɛ j (t) 2] + 2 dt j Z j Z(ɛ 2 j (t) ɛ j (t)ɛ j 1 (t)) = O j (t)ɛ j (t). (5) j Z }{{} = 1 (ɛ 2 j (t) ɛ j 1 (t)) 2 0 j Z ɛ (t) 2 l 2 := ɛ j (t) 2. By Caucy-Scwarz inequality in (5) 1 d 2 dt ɛ (t) 2 l j Z 2 O (t) l 2 ɛ (t) l 2, so tat d dt ɛ (t) l 2 O (t) l 2 and, since ɛ (0) l 2 = 0, Assume f C 2 c (R). Ten O j (t) = f (x j 1/2 t) /2 and Teorem O (s) 2 l 2 = j Z f (x j 1/2 s) 2 = 3 4 Supp(f ) t ɛ (t) l 2 0 O (s) l 2 ds. j Z s.t. x j 1/2 s Suppf f (x j 1/2 s) 2 f 2 L (R) = 2 4 Supp(f ) f 2 L (R). For any initial data f C 2 c (R) in te transport equation, te backward semi-discrete sceme wit initial data u j (0) = f (x j ) is convergent of order in l 2 and te error ɛ j (t) = u j (t) u(x j, t) satisfies te estimate ɛ (t) l 2 t 2 f L (R) Supp(f ), t 0, > 0. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

12 More about te energy metod Te L 2 (R)-norm of te solution for te continuous transport equation ut + ux = 0 is conserved in time. Conservation law of te energy: d dt u(, t) 2 L 2 (R) = 0. Te centered semi-discrete sceme is also conservative: d dt u (t) 2 l 2 = 0. Te backward semi-discrete sceme is dissipative since te energy decreases in time: d dt u (t) 2 l 2 +, u (t) 2 l 2 = 0, were, f j := f j f j 1. Te forward semi-discrete sceme is anti-dissipative since te energy increases in time: d dt u (t) 2 l 2,+ u (t) 2 l 2 = 0, were,+ f j := f j+1 f j. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

13 Fully discrete scemes for te transport equation Consistency exercise Leap-frog sceme: u k+1 j u k 1 j + uk j+1 uk j 1 = 0. 2dt 2dx Stability von Neumann metod. Set û,k (ξ) - te semi-discrete Fourier transform at scale of te solution at time t k, (u k j ) j and µ := dt/dx - te Courant number. Te sequence (û,k (ξ)) k verifies te second-order recurrence: û,k+1 (ξ) + 2iµ sin(ξ)û,k (ξ) û,k 1 (ξ) = 0. Te two roots of te caracteristic polynomial are: λ ± (ξ) = iµ sin(ξ) ± 1 µ 2 sin 2 (ξ). Wen µ < 1, 1 µ 2 sin 2 (ξ) > 0 for all ξ, so tat λ ± (ξ) C of te same imaginary part and of opposite real parts. Also λ ± (ξ) 2 = µ 2 sin 2 (ξ) + 1 µ 2 sin 2 (ξ) = 1. Te stability is guaranteed by te fact tat bot roots λ ± (ξ) are simple and of modulus 1, for any ξ. Wen µ = 1, 1 µ 2 sin 2 (ξ) > 0, excepting te case ξ = π/2 and ξ = 3π/2, for wic tere is a double root of unit modulus INSTABILITY. Wen µ > 1, tere exists ξ µ (0, 2π/) s.t. 1 µ 2 sin 2 (ξ) > 0 for all ξ (0, ξ µ) and 1 µ 2 sin 2 (ξ) 0 for all ξ [ξ µ, 2π). In tis last case, te metod is UNSTABLE: λ ± (ξ) = i(µ sin(ξ) µ 2 sin 2 (ξ) 1) and λ (ξ) = µ sin(ξ) + µ 2 sin 2 (ξ) 1 > 1. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

14 Fully discrete scemes for te transport equation backward Euler EXPLICIT: u k+1 j u k j dt + uk j u k j 1 dx Stability von Neumann metod. Te sequence (û,k (ξ)) k verifies te first-order recurrence: û,k+1 (ξ) = [1 + µ(exp(iξ) 1)]û,k (ξ). For te stability is sufficient to guarantee tat 1 + µ(exp(iξ) 1) 1: 1 + µ(exp(iξ) 1) 2 = (1 + µ(cos(ξ) 1)) 2 + µ 2 sin 2 (ξ) = 1 + 2µ(µ 1)(1 cos(ξ)) 1 iff µ 1. = 0. backward Euler IMPLICIT: u k+1 j u k j dt + uk+1 j u k+1 j 1 = 0. dx Stability von Neumann metod. Te sequence (û,k (ξ)) k verifies te first-order recurrence: û,k+1 (ξ) = µ(1 exp(iξ)) û,k (ξ). For te stability is sufficient to guarantee tat 1 + µ(1 exp(iξ)) 1: 1 + µ(1 exp(iξ)) 2 = (1 + µ(1 cos(ξ))) 2 + µ 2 sin 2 (ξ) = 1 + 2µ(µ + 1)(1 cos(ξ)) 1, µ > 0 UNCONDITIONAL STABILITY NO CONDITION on µ to guarantee stability. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

15 Fully discrete scemes for te transport equation Teorem If û,k+1 (ξ, t) û,k (ξ) for all ξ [ π/, π/] u,k+1 l 2 u,k l 2. Proof: Parseval identity for te SDFT. Oter fully discrete scemes for te transport equation: Crank-Nicolson, inspired from te trapezoidal rule for solving ODEs, is unconditionally stable and of second-order in bot time and space: u k+1 j uj k dt [ u k+1 j+1 uk+1 j 1 2dx + uk j+1 uk ] j 1 = 0. 2dx Lax-Wendroff, of second-order, conservative, stable iff µ 1 (exercise) u k+1 j uj k dt + uk j+1 uk j 1 2dx Lax-Friedrics, of first-order, stable iff µ 1 (exercise) u k+1 j 1 2 (uk j+1 + uk j 1 ) dt dt uj+1 k 2uk j + uj 1 k 2 dx 2 = 0 + uk j+1 uk j 1 2dx = 0. Definition (A-stability, cf. Iserles, A first course on numerical analysis of ODEs) A numerical metod is A-stable if it preserves te beaviour of te continuous solution as t. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

16 Numerical approximations for te wave equation Te finite difference space semi-discretization: u j u j+1 2u j +u j 1 2 = 0. Te explicit leapfrog fully discrete finite difference sceme is stable for µ = dt/dx 1: u k+1 2u k j j +uk 1 j dt 2 uk j+1 2uk j +uk j 1 dx 2 = 0. Te implicit leapfrog fully discrete finite difference sceme is unconditionally stable: u k+1 2u k j j +uk 1 j dt 2 u k+1 j+1 2uk+1 +u k+1 j j 1 dx 2 = 0. Te implicit midpoint sceme is unconditionally stable: u k+1 2u k j j +uk 1 j dt u k+1 j+1 2uk+1 +u k+1 j j 1 dx u k 1 j+1 2uk 1 +u k 1 j j 1 dx 2 = 0. Te finite element semi-discretization. Find u (x, t) = N u j (t)φ j (x) V := span{φ 1,, φ N } s.t. j=1 d 2 1 dt 2 u 1 (x, t)φ(x) dx + ux (x, t)φ x (x) dx = 0, φ V. 0 0 x x j 1, x (x j 1, x j ) Here φ j (x) = x j+1 x, x (x j, x j+1 ),, Ten (u j (t)) j satisfies te system: 0, oterwise. 6 u j+1 (t) u j (t) + 6 u j 1 (t) u j+1(t) 2u j (t)+u j 1 (t) = 0. Finite difference semi-discretization of te 2 d wave equation: u j,k (t) u j+1,k (t) 2u j,k (t)+u j 1,k (t) x 2 u j,k+1(t) 2u j,k (t)+u j,k 1 (t) y 2 = 0. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

17 Numerical dissipation and dispersion Trefeten [6]: Finite difference approximations ave more complicated pysics tan te equations tey are designed to simulate. Tey are used not because te numbers tey generate ave simpler properties, but because tose numbers are simpler to compute. Plane wave solutions: u(x, t) = exp(i(ξx + tω)), were ξ is te wave number and ω is te frequency. Te PDE or te numerical sceme imposes a relationsip between ω and ξ, ω = ω(ξ), called dispersion relation. Examples: Transport equation u t + u x = 0 ω(ξ) = ξ Wave equation u tt u xx = 0 ω(ξ) = ±ξ Scrödinger equation iu t + u xx = 0 ω(ξ) = ξ 2. Centered finite difference semi-discretization for te transport equation u j + (u j+1 u j 1 )/2 = 0 ω (ξ) = sin(ξ)/ Finite difference semi-discretization of te wave equation u j (u j+1 2u j + u j 1 )/ 2 = 0 ω (ξ) = ±2 sin(ξ/2)/. Finite difference semi-discretization of te Scrödinger equation iu j + (u j+1 2u j + u j 1 )/ 2 = 0 ω (ξ) = 4 sin 2 (ξ/2)/ 2. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

18 Dispersion relations for fully discrete scemes for te transport equation Leap-frog: sin(dtω (ξ)) + µ sin(dxξ) = 0, µ = dt/dx, = dx. Backward explicit Euler: exp(idtω (ξ)) 1 + µ(1 exp( iξdx)) = 0 Backward implicit Euler: 1 exp( idtω (ξ)) + µ(1 exp( iξdx)) = 0 Crank-Nicolson: 2 tan(dtω (ξ)/2) + µ sin(dxξ) = 0 Lax-Wendroff: exp(idtω (ξ)) 1 + iµ sin(dxξ) + 2µ 2 sin 2 (dxξ/2) = 0 Lax-Friedric: exp(idtω (ξ)) cos(dxξ) + iµ sin(dxξ) = 0. Definition A finite difference sceme is dissipative of order 2r if te dispersion relation satisfies Im(ω (ξ)dt) γ ξdx 2r, for all ξ [ π/dx, π/dx], γ > 0. A finite difference sceme is non-dissipative if Im(ω (ξ)) = 0. Example non-dissipative scemes: leap-frog, Crank-Nicolson Example dissipative scemes: Lax-Wendroff, backward explicit Euler, Lax-Friedric (in figure)... Effect of dissipation: Te amplitude of te numerical solution decays in time. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

19 Group velocity, pase velocity Wave packet: u(x, t) = R φ(ξ) exp(itω(ξ) + iξx) dξ. Data concentrated around x = 0 and oscillating at frequency ξ 0 : φ(x) = ψ(x) exp(iξ 0 x). Example - finite difference semi-discretization: ω = ω (ξ) = 2 sin(ξ/2)/ ω (ξ 0 ) + (ξ ξ 0 )ω (ξ0 ) u(x, t) exp(iξ 0 (x + tω (ξ 0 )/ξ 0 ))ψ(x + tω (ξ0 )). Pase velocity: ω (ξ 0 )/ξ 0 - te velocity of propagation for te oscillation Group velocity: ω (ξ0 ) - te velocity of propagation for te envelope ψ. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

20 Geometric Optics Linear transport equation Ligt as a dual nature: is particle (poton) and is wave (and oscillates at a certain wavelengt). Geometric Optics (GO) studies te propagation of ligt particles along trajectories called rays. Hamilton principle states tat te trajectory of a particle between times t 1 and t 2 minimizes te t 2 action L(q, q, t) dt, were L = T V is te difference between kinetic and potential energies, t 1 q is te vector of generalized coordinates and q = tq. Hamiltonian system associated to H = H(p, q): p (s) = qh(p(s), q(s)), q (s) = ph(p(s), q(s)). For te continuous wave equation, H = H(x, t, ξ, τ) = τ 2 ξ 2 and te rays of GO verify te Hamiltonian system: x (s) = ξ H(x(s), t(s), ξ(s), τ(s)) = 2ξ(s), t (s) = τ H(x(s), t(s), ξ(s), τ(s)) = 2τ(s) ξ (s) = x H(x(s), t(s), ξ(s), τ(s)) = 0 ξ(s) = ξ 0, τ (s) = th(x(s), t(s), ξ(s), τ(s)) = 0 τ(s) = τ 0. Null-bi-caracteristics. H(x(0), t(0), ξ(0), τ(0)) = 0. Ten H(x(s), t(s), ξ(s), τ(s)) = 0 s > 0. Caracteristics. Replace s by t in te Hamiltonian system. Since H(x(0), 0, ξ(0), τ(0)) = 0 as two roots as equation in τ 0, τ 0 = ± ξ 0, ten x (t) = ±ξ 0 / ξ 0 two caracteristics: x(t) = x 0 ± tξ 0 / ξ 0. Tey propagate at unit velocity. For te finite difference semi-discretization of te wave equation, te Hamiltonian is H(x, t, ξ, τ) = τ 2 4 sin 2 (ξ)/ 2 and te caracteristics propagate wit te group velocity, i.e. x(t) = x 0 ± t cos(ξ 0 /2) (exercise). (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21

21 Some related bibliograpy T.-C. Poon, Engineering Optics Wit Matlab, World Scientific, A. Quarteroni, A. Valli, Numerical approximation of PDEs, Springer, J.C. Strikwerda, Finite difference scemes and PDEs, SIAM, L.N. Trefeten, Group velocity ub finite difference scemes, SIAM Rev L.N. Trefeten, Spectral metods in Matlab, SIAM, L.N. Trefeten, Finite difference and spectral metods for ordinary and PDEs, R. Vicnevetsky, B. Bowles, Fourier analysis of numerical approximations of yperbolic equations, SIAM, E. Zuazua, Métodos numéricos de resolución de Ecuaciones en Derivadas Parciales, capters 3 and 9. (BCAM - Basque Center for Applied Matematics) Finite difference approximations Bilbao /06/ / 21