Laurette TUCKERMAN Numerical Methods for Differential Equations in Physics

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1 Laurette TUCKERMAN Numerical Methods for Differential Equations in Physics

2 Time stepping: Steady state solving: 0 = F(U) t U = LU + N(U) 0 = LU + N(U) Newton s method 0 = F(U u) F(U) DF(U)u { DF(U)u = F(U) U U u

3 Newton s Method converges quadratically U n+1 = U n F(U n) F (U n ) F(U) = 0 = F(U n ) + F (U n )(U U n ) F (U n )(U U n ) = F(U n) F (U n ) + F (U n ) F (U n ) (U U n) + 1 F (U n ) 2 F (U n ) (U U n) = F(U n) F (U n ) + (U U n) = U n+1 + U U n+1 U = 1 F (U n ) 2 F (U n ) (U U n) ǫ n+1 = 1 2 F (U n ) F (U n ) ǫ F (U n ) F (U n ) (U U n) F (U n ) F (U n ) (U U n) Typical sequence: ǫ = 10 1,10 2,10 4,10 8, 10 16

4 Much faster than timestepping: U(t) = U + ce λt U(t n ) U = ce λt n U(t n+1 ) U = ce λ(tn+ t) = e λ t (U(t n ) U)) Linear convergence: ǫ n+1 cǫ n. In addition to converging faster than timestepping, Newton s method can converge to unstable states.

5 Fixed points and linear stability. ẋ = f(x) unstable stable 0 = f( x) Fixed point x d ( x + ǫ(t)) = f( x + ǫ) dt Linear stability of x ǫ = f( x) + f ( x)ǫ f ( x)ǫ 2 + f ( x)ǫ { increases iff ǫ(t) = e tf ( x) ǫ(0) ( x) > 0 decreases iff ( x) < 0

6 Saddle-node Bifurcations ẋ = f(x) = µ x 2 Fixed points: Stability: x ± = ± µ forµ > 0 f ( x ± ) = 2 x ± = 2(± µ) = 2 µ f ( x + ) = f ( µ) = 2 µ < 0 = x + stable f ( x ) = f ( µ) = 2 µ > 0 = x unstable

7 f(x,µ) = c 00 + c 10 x + c 01 µ + c 20 x general quadratic polynomial = ± µ ± x 2 four cases, depending on signs ofc s Newton s method finds steady states independently of their stability Where might saddle-node bifurcations occur?

8 Swift-Hohenberg equation t u = µu ( q 2 c + ) 2 u u 3 Derived by J. Swift and P.C. Hohenberg (Phys. Rev. A 15, 319 (1977)) to describe pattern formation in convection For u 1, t u = µu ( q 2 c + ) 2 u u exp(ikx + σt) σu = µu ( q 2 c k2) 2 u = σ = µ,k = qc Add quadratic term to obtain hexagons t u = µu ( q 2 c + ) 2 u + g1 u 2 u 3 Include q c and q c = 1 to obtain quasipatterns t u = µu ( q 2 c + ) 2 (1 + ) 2 u + g 1 u 2 u 3

9 2D Patterns produced by Swift-Hohenberg equation Stripes Hexagons Zigzag instability Quasicrystals

10 Snaking in 1D Swift-Hohenberg Equation

11 Thermosolutal Convection: Patterns with 1D snaking

12 Thermosolutal Convection: Patterns with 2D snaking

13 Newton s method: example Swift-Hohenberg equation: t U = F(U) = µu ( q 2 c + ) 2 U U 3 Equation for steady state: 0 = F(U) = µu ( q 2 c + ) 2 U U 3 Loop: calculate and compare withǫ: F(U) µu ( q 2 c + ) 2 U U 3 < ǫ? If F(U) <ǫ, thenu not solution, so tryu u: 0 = µ(u u) ( q 2 c + ) 2 (U u) (U u) 3 = µu ( q 2 c + ) 2 U U 3 (µu ( ) q 2c + ) 2 u 3U 2 u 3Uu 2 u 3 Newton step: truncate at first order in u and solve foru(x): µu ( q 2 c + ) 2 u 3U 2 u = µu ( q 2 c + ) 2 U U 3 Then replace and try again: U U u

14 Continuation: going around saddle-nodes

15 Goal: 0 = RN(U) + LU 0 = p(u,r) p where Newton step: { Ui some component R } (U,R) not solution, so try (U u,r r) 0 = (R r)n(u u) + L(U u) = RN(U) + LU RN U u rn(u) Lu + O(r,u) 2 { } Ui p u 0 = p(u u,r r) p = i R p r

16 [ RN U + L ] N(U) }{{ 1 } or [ u r ] = RN(U) { + LU } Ui p R p Ifp(U,R) = R (i.e. set Reynolds number), then setr = p, r = 0 and get previous case: [RN U + L][u] = [RN(U) + LU] Ifp(U,R) = U i, then must solve extended system for(u,r). [ (RN u + L) N(U) ][ u r ] = Setu i = U i p Calculate (RN U + L)u AddN(U)r [ (RN(U) + LU) U i p ]

17 It may be sufficient to extrapolate quadratically. Far from saddle-node bifurcation,u is considered to be a function ofr. To get an initial guess foru at a newr, extrapolate. Zeroth order extrapolation: Linear extrapolation: U(R (2) ) initial guess = U(R (1) ) U(R (2) ) initial guess = U(R (1) ) + (U(R (1) ) U(R (0) )) R(2) R (1) R (1) R (0) Quadratic extrapolation: Fit quadratic polynomials through U(R (0) ), U(R (1) ), U(R (2) ) as functions ofrand evaluate polynomial at new value R (3). Close to saddle-node bifurcation, choose distinguished value U i and consider U j,j i and R to be quadratic functions of U i. Set new value ofu i, and evaluate new estimate of U j and R Now, R can change sign and can go around saddle-node.

18 Reaction-Diffusion Equations t u i = f i (u 1,u 2,...) + D }{{}} i{{ u } i reaction diffusion Reactions f i couple different speciesu i at same location DiffusivityD i couples same speciesu i at different locations Describe oscillating chemical reactions, such as famous Belousov-Zhabotinskii reaction, discovered by two Soviet scientists in 1950s-1960s. Also describe phenomena in biology (population biology, epidemiology, neurosciences) social sciences (economics, demography) physics

19 Two species t u = f(u,v) + D u u t v = g(u,v) + D v v FitzHugh-Nagumo model f(u,v) = u u 3 /3 v + I g(u,v) = 0.08 (u v) Spatially homogeneous t u = f(u,v) t v = g(u,v) Barkley model f(u,v) = 1 ǫ u(1 u)( u v+b ) a g(u,v) = u v u-nullclinesf(u,v) = ( 0, v-nullclinesg(u,v) ) = 0, steady states fu f stable if eigenvalues of v have negative real parts g u g v

20 f(u,v) = 1 ǫ u(1 u)( u v+b ) a g(u,v) = u v t u = f = 0 separates and O(ǫ 1 ) t v = g = 0 separates and O(1) u = 1 u = 0 u = 0 u = (v + b)/a excited phase v 1 refractory phase v 1 excitable phase excitation threshold

21 Waves in Excitable Medium Spatial variation + diffusion + excitability= propagating waves Excitable media in physiology: neurons cardiac tissue (the heart) Pacemaker periodically emits electrical signals, propagated to rest of heart

22 Simulations from Barkley model, Scholarpedia Spiral waves in 2D Spiral waves in 3D

23 TRAVELING WAVES: U(x Ct,y,z) Newton step: { 0 = C x U + N(U) + LU Goal : 0 = p(u) p (U,C) not solution, so try(u u,c c) 0 = (C c) x (U u) + N(U u) + L(U u) = C x U + N(U) + LU C x u c x U N U u Lu { } Ui p u 0 = p(u u,r r) p = i R p r [ C x + N U + L x U ][ u c ] = [ C x U + N(U) + LU U i p ]

24 Navier-Stokes Equations t U = (U )U P + ν U = (I 2 )(U )U + ν U = N(U) + L U N U u (U )u (u )U A U u = N U u + Lu Must solve Lu + N U u = LU + N(U)

VII. Reaction-Diffusion Equations:

VII. Reaction-Diffusion Equations: Université Pierre et Marie Curie Master Sciences et Technologie (M) Spécialité : Concepts fondamentaux de la physique Parcours : Physique des Liquides et Matière Molle Cours : Dynamique Non-Linéaire Laurette