Numerical Evans Function Computation Jeffrey Humpherys Brigham Young University INTRODUCTION

Transcription

1 Numerical Evans Function Computation Jeffrey Humpherys Brigham Young University INTRODUCTION 1

2 Traveling Wave Consider the evolution equation u t = F(u, u x,u xx,...). A traveling wave with speed is a solution u(x, t) =û(x st). Or equivalently a stationary solution of u t su x = F(u, u x,u xx,...). s Existence of Profile A front is a heteroclinic point A pulse is a homoclinic point A wavetrain is a periodic solution 2

3 Nonlinear Stability Examine flow near the stationary solution u(x, 0) = û(x)+v(x, 0), where kv(x, 0)k is small. Asymptotically orbital stability when lim u(x, t) =û(x + δ). t Example Isentropic Navier Stokes v t u x =0 u t + p(v) = (play movie) ³ ux v x 3

4 Spectral Stability Linearization about the profile v t =(df(û) s x )v + Q(v, v x,v xx,...). {z } {z } Lv Higher order Spectral Stability when σ(l) P =, where P = {λ <e(λ) 0}\{0}. EVANS FUNCTION 4

5 Eigenvalue Problem Write the eigenvalue problem λv = Lv, <x<, as a first-order system Assume: ( W 0 = A(x, λ)w, W(± ) =0. W C n A(x, λ) is consistently split and asymptotically constant in x. Define Evans Function D(λ) =W W + k {z } S + (λ) {W + i }k i=1 {W j }n j=k+1 W k+1... W n {z } U (λ) where and are analytic bases of the stable/unstable manifolds of x = ±, respectively. 5

6 Problems with Stiffness {W + i }k i=1 Maintaining linear independence of and {W j }n j=k+1 is hard. Common Options: Orthogonalization bad for winding numbers due to loss of analyticity. Compound-Matrix Method numerically prohibitive for moderate values of n. COMPOUND-MATRIX METHOD 6

7 Compound-Matrix Method I Developed by Gilbert & Bakkus (1966) and Ng & Reid (1979). First used to numerically compute Evans functions by Pego (1995). Use in Evans functions rediscovered and further developed by Brin & Zumbrun (1998) and Bridges & collaborators (2002). Compound Matrix Method II Lift to wedge-product space W 0 = A (k) (x, λ)w, W C (n k). Flow of smallest simple mode at x = corresponds to the k-form spanning the stable manifold. Similar approach holds for x =. 7

8 Lifting to wedge-product space I Let {w i } n i=1 be a basis for C n. Then {w i1... w ik i 1 <...<i k } is a basis for Λ k (C n ). Hence A : C n C n induces a linear map A (k) : Λ k (C n ) Λ k (C n ) satisfying Lifting to wedge-product space II A (k) w i1... w ik kx = w i1... Aw ij... w ik. j=1 8

9 Lifting to wedge-product space III If in an eigenbasis Aw i = µ i w i, then A (k) w i1... w ik kx = w i1... Aw ij... w ik j=1 = kx j=1 µ ij w i1... w ik. Example Good Boussinesq equation u tt = u xx u xxxx (u 2 ) xx, admits the profile ū(ξ) = 3 2 (1 s2 )sech 2 ³ 1 s 2 2 ξ which is known to be stable when and unstable when s < 1 2., ξ = x st, 1 2 s < 1 9

10 Example (cont) Linearization about the profile yields hence λ 2 u 2sλu 0 =(1 s 2 )u 00 u 0000 (2ūu) 00, u A(x, λ) = , W = u 0 u 00. λ 2 2ū xx 2λs 4ū x (1 s 2 ) 2ū 0 u 000 Example (cont) Lifting to wedge-product space yields A (2) (x, λ) = 2λs 4ū x (1 s 2 ) 2ū λ 2 +2ū xx 0 0 (1 s 2 ) 2ū λ 2 +2ū xx 0 2λs +4ū x

11 Compound-Matrix Blow-Up Note: Matrix-vector multiplication scales as Note: n 2. k 8 4 =70, 14 7 = 3432, =184, 756. Sparse matrix-vector multiplication scales as (k(n k)+1) n k. POLAR-COORDINATE METHOD 11

12 Evans Function Reformulation I Let and then for a k k matrix α + and a (n k) (n k) matrix α, we have and where W + =(W W+ k ) W =(W k+1...w n ) Ω H + Ω + = I k and W = Ω α Ω H Ω = I n k. W + = Ω + α + Evans Function Reformulation II Then and W + 1 W + k =(detα +) {z } γ + (Ω + 1 Ω+ k ) W k+1 W n =(detα ) {z } γ (Ω k+1 Ω n ). Hence, we define the Evans function to be D(λ) =det(w +,W ) x=0 = γ + γ det (Ω +, Ω ) x=0. 12

13 Continuous Orthogonalization Choose k k such that matrix-valued function α(x) W (x) =Ω(x)α(x), W,Ω C n k, Ω H Ω = I k. Then W 0 = A(x, λ)w becomes Ω 0 (x) =A(x, λ)ω(x)+ω(x) α 0 (x)α(x) 1. {z } g(x) Drury Method I Let then g = Ω H AΩ Ω 0 = AΩ ΩΩ H AΩ =(I ΩΩ H )AΩ. 13

14 Drury Method II Invariant on Stiefel Manifold or V k (C) ={Ω C n k Ω H Ω = I k } where E 1 (0) E(Ω) :=Ω H Ω I k. Numerically unstable E 0 = g H E + Eg. Davey Method I Let then g = (Ω H Ω) 1 Ω H AΩ {z } Ω Ω 0 = AΩ ΩΩ AΩ =(I ΩΩ )AΩ. 14

15 Davey Method II Numerically stable E 0 =0. But still not attracting on the Stiefel manifold. One variation is to add a damping term Ω 0 =(I ΩΩ )AΩ + γω(i Ω H Ω), then E 0 = 2γ(I + E)E. Polar Coordinate Method I Note AΩα = Ω 0 + Ωα 0. We can left multiply by Ω to get since α 0 = Ω AΩα Ω Ω 0 = Ω (I ΩΩ )AΩ =0. 15

16 Polar Coordinate Method II By Abel s theorem, we get γ 0 =tr(ω AΩ)γ, γ =detα. All combined, we have ( Ω 0 =(I ΩΩ )AΩ γ 0 =tr(ω AΩ)γ Shooting on both ends yields D(λ) =γ + γ det (Ω +, Ω ) x=0. Operational Count With damping the polar-coordinate method grows as kn 2 +4k 2 n. With sparse solver, the compound-matrix method grows as (k(n k)+1) n k. Break even between n=7 and n=8. 16

17 OTHER NUMERICAL ISSUES Other Issues Kato s Method Integrated Coordinates Geometric Integrators Contours how big is big enough? 17

18 EXAMPLES 2.5 x Good Boussinesq (revisited) (a) We compute the Evans function along real axis from λ =0toλ =0.2 18

19 x Good Boussinesq (re-revisited) (b) x 10 5 TheimageoftheclosedcontourΓ(t) = e 2πit. Isentropic Navier Stokes (show Matlab figures) 19

20 Future Directions Push the limits of the method. Test different continuous orthogonalization methods (e.g., geometric integrators). Semi-spectral methods instead of shooting? Further development of STABLAB. 20