and Diffusion-Wave Problems

Transcription

1 Adaptive Finite Difference Method with Variable Timesteps for Fractional Diffusion and Diffusion-Wave Problems Santos B. Yuste and Joaquín Quintana-Murillo Dpt. Física, Univ. Extremadura Badajoz (Spain)

2 The problem in a nutshell For some anomalous diffusive (subdiffusive) particles, the pdf of finding them at a given place at a given time follows a fractional (i.e., an integro-differential ) diffusion equation. When one solves this equation by means of standard finite difference methods, the CPU time and computer memory consumption scale as time 2!!! Solution from t= 0 to t =1 Solution from t= 0 to t = = time CPU time to reach t=1 >> CPU time to reach t=1 Standard method Our adaptive method 4

3 Brownian motion. Normal diffusion Fig. from the web page of this Symposium Recorded random walk trajectories by Jean Baptiste Perrin. Lef part: three designs obtained by tracing a small grain at intervals of 30 s. Right part: the starting point of each motion event is shifted to the origin. These figures constitute part of the measurement of Perrin, Dabrowski and Chaudesaigues leading to the determination of the Avogadro number. The result was R. Metzler, J. Klafter / Physics Reports 339 (2000) 1-77 Jean Baptiste Perrin Green function u(x,t): concentration of particles at x at @2 u(x; t) = D u(x; u(x; t) = 1 p 4¼Dt e x2 =4Dt Diffusion equation Green function 6

4 Subdiffusion in... Physics Geology Finance Chemistry Ecology. Biology (see next) (anomalous) diffusion exponent in vivo: γ =0.7 Golding and Cox Physical Nature of Bacterial Cytoplasm PRL 96, (2006) 8

5 How can we model subdiffusion processes?? 10

6 CTRW with fat/heavy tail Subdiffusion Non-Gaussian Green function P(x,t) CTRW Jump distribution (x) with GLCT finite variance and waiting-time distribution ψ(t) with power-law Subdiffusion: 0< u(~r; t) = Kr2 u(~r; t) Fractional diffusion equation 12

7 Subdiffusion Fractional diffusion equation u(x; t) = K y(t) 1 (1 ) Z t 0 d 1 (t ) dy( ) d ; 0 < < 1 Limit equation of a (mesoscopic) CTRW model with powerlaw distribution of waiting times Linear Easy inclusion of external fields and boundary conditions analytical techniques and solutions for some basic problems (eigenfunction expansions, Green functions) algorithms for obtaining numerical solutions

8 Finite difference method: numerical u(x; t) = F(x; 2 ±U = F ± t K± 2 x U n j = F(x j ; t n ) (1) ± t U n j = 1 (2 ) n 1 X m=0 T m;n ( ) U m+1 j Uj m (2) (3) ± xu(x 2 j ; t) = u(x j+1; t) 2u(x j ; t) + u(x j 1 ; t) ( x) 2 S n U n j+1 + (1 + 2S n )U n j S n U n j 1 = MU n j + ~ F(x j ; t n ) MU n j U n 1 j n 2 X m=0 ~T m;n ( ) U m+1 j U m j S n = (2 )K (t n t n 1 ) ( x) 2 ;

9 Fractional difference methods are heavy t 0 t m 2 t m 1 t m t m+1 U0 j 1; Um 2 U m 1 Uj 1 m j 1 ; j 1 ; U m+1 j 1 U 0 j ; U m 2 j ; Uj m 1 ; Uj m U m+1 j U0 j+1; Um 2 U m 1 Uj+1 m j+1 ; j+1 ; U m+1 j+1 m terms From m to m+1: computational cost m nx From m=1 to m=n+1: computational cost m 2» n 2 m=1 Huge!!! [ computational cost for normal diffusion n ]

10 Variable/adaptive timestesps: Adaptive methods u t m-1 t m t m+1 Small timesteps t Large timesteps Adaptive methods more reliable: via thorough sampling of difficult regions faster : via sparse sampling of quiet regions A good ODE integrator should exert some adaptive control over its own progress, making frequent changes in its stepsize. [ ] Many small steps should tiptoe through treacherous terrain, while a few great strides should speed through smooth uninteresting countryside. The resulting gains in efficiency are not mere tens of percents or factors of two; they can sometimes be factors of ten, a hundred, or more 22 Press et al, Numerical Recipes, section 16.2

11 A testbed problem with wild and quiet regions Dispersion of a constant flux of subdiffusive particles arising from a point source ( one particle released every unit time at the source) t =1 + t 0+ t=1 + t 1 + as times goes by and so on

12 A testbed problem with wild and quiet regions Dispersion of a constant flux of subdiffusive particles one particle released every unit time at x=0 : 1D infinite medium A particle appears at x=0 x=0 ψ(t) ψ(t) Source of particles t=0 x=0 ψ(t) t m A second particle appears at x=0 ψ(t) x=0 ψ(t) t n t=1 t p The goal: u(x,t) = density of probability of finding a particle at x at time t 24

13 Formation of morphogen gradients: standard reaction-diffusion model diffusion random walk Source of particles Source of particles 1D testbed problem + Particles (morphogens) may react and disappear development of morphogens gradients Long-time (stationary) distribution of morphogens in the medium is especially important Fast and accurate numerical methods are required

14 A testbed problem with wild and quiet regions The release of a particle at x=0 is described by means of a Dirac delta function Exact solution u(0,t) vs t u(0,t) u(0,t) t Dirac delta function n a new particle appears at t=n : u(x,n + )=u(x,n - )+δ(x) 37

15 Discretization of δ(x) Dirac delta function approximated by a hat function δ(x) 1/ x Discretized numerical approximation to the Dirac s delta initial condition: i x u(x j ; 0) = ½ 1= x; j = 0 0; j 6= 0

16 Choosing the timesteps g(t m ) = (U m 1 2U m 0 + U m 1 )=( x) 2 2 u(x; t m )=@x 2 j x=0 curvature at the origin Size of the timesteps t m+1 t m = min 10 4 coth [jg(t m )j=1000] ; 0:02 t m+1 t m Out[40]= g curvature Arbitrary but with convenient features: Timestep decreases (increases) when the curvature increases (decreases) Prefixed maximum and minimum values (0.02 and , respectively)

17 10-3 = time Standard method u(x,t) t=0 + Adaptive method x u(x,t) t= timestep 10 timesteps 3.0 t= x t= timesteps u(x,t) t= x 66 timesteps u(x,t) t=1-65 timesteps x t=2-141 timesteps u(x,t) u(x,t) u(x,t) x x x

18 Solution at the most difficult place, x=0, where the particles are born Tiny timestep u(0,t) u(0,t) Large timestep t u(0,t) t Medium timestep t Solid symbols: adaptive method Hollow symbols: method with fixed timestep: t m+1 -t m =0.001 Line: exact solution

19 Error at the most difficult place, x=0, where the particles are born 1 δ Dirac delta function approximated by a hat function 0.1 e(0,t) Dirac delta function approximated by a hat function δ 1E t e(0,t) 0.01 Solid symbols: adaptive method Hollow symbols: method with fixed timestep: t m+1 -t m = E t

20 Another testbed problem. Another adaptive algorithm A problem with initial condition easier to handle 1 u(x,0) u(x,t)=? = 2 0 x ¼ u(x = 0; t) = u(x = ¼; t) = 0 u(x; t) = E ( t ) sin(x) u(x; 0) = sin x K = 1 46

21 How to choose the size of the timesteps? Another adaptive algorithm t m+1 t m = min 10 4 coth [jg(t m )j=1000] ; 0:02 Out[40]= Previous criterium: taylored for the problem at hand Goal: a more general criterium. A good ODE integrator should exert some adaptive control over its own progress, making frequent changes in its stepsize. Usually the purpose of this adaptive stepsize control is to achieve some predetermined accuracy in the solution with minimum computational effort. Many small steps should tiptoe through treacherous terrain, while a few great strides should speed through smooth uninteresting countryside. Prest et al. Numerical recipes. We explore an adaptive algorithm of classic style 48

22 Another adaptive algorithm We explore a straightforward algorithm of classic style NUMERICAL RECIPES IN FORTRAN 77: THE ART OF SCIENTIFIC COMPUTING Section 16.2: Usually the purpose of this adaptive stepsize control is to achieve some predetermined accuracy in the solution with minimum computational effort. Many small steps should tiptoe through treacherous terrain, while a few great strides should speed through smooth uninteresting countryside. [ ] Implementation of adaptive stepsize control requires that the stepping algorithm return information about its performance, most important, an estimate of its truncation error. [ ] Obviously, the calculation of this information will add to the computational overhead, but the investment will generally be repaid handsomely With fourth-order Runge-Kutta, the most straightforward technique by far is step doubling. We take each step twice, once as a full step, then, independently, as two half steps. The difference between the two numerical estimates is a convenient indicator of truncation error. It is this difference that we shall endeavor to keep to a desired degree of accuracy, neither too large nor too small. We do this by adjusting [the timestep]

23 How to choose the size of the timesteps? A new algorithm Step 2a met. accurate Step 2b met. fast and still accurate? 1. Boostrap of step n: n = n 1 and (n) bu k U (n) k > tolerance (n) 2a. True: then until bu k U (n) n! n =R k < then t n = t n 1 + n 2b. False: then n! R n until U b(n) k 3. Repeat [i.e., n! n + 1 and go to 1] U (n) k > then t n = t n 1 + n =R R: brave exploration coefficient we use R=2 52

24 Step doubling algorithm Numerical results γ=1/2 Solid line: exact solution; squares: adaptive numerical solution up to n = 45 (t 45 = 9:6268) and tolerance = ; stars: numerical solution up to n = 112 (t 112 = 9:1690) and = In both cases 0 = 0:01 and x = ¼=40.

25 Step doubling algorithm Numerical results γ=1/2 Errors in the adaptive algorithm are even (great!, compare with those for fixed timesteps) Numerical errors at the mid-point of (i) the adaptive method with = , 0 = 0:01 (squares, 45 timesteps, CPU time ¼ 0:15s ); (ii) the adaptive method with = 10 4 and 0 = 0:01 (stars, 112 timesteps, CPU time ¼ 0:75s); (iii) the method with constant timesteps of size n = 0:01 (triangles, 1010 timesteps, CPU time ¼ 440s); and (iv) the method with uniform timesteps of size n = 0:001 (circles, timesteps, CPU time ¼ 43800s). In all cases = 1=2 and x = ¼=40.

26 Does the method always work? Stability Do the inevitable numerical perturbations (numerical noise, round-off errors) ruin the method? How do the perturbations evolve? v (m) j bu (m) j U (m) j = v (m) j unstable perturbed solution perturbation 1 stable 2 m Linear AU (n) = MU (n) + ~F (n) Av (n) = Mv (n) Eq. for the evolution of the perturbation

27 Does the method always work? Stability Von Neumann stability analysis: (Yuste-Acedo, arxiv:cs/ , SIAM J. Num. Anal. 05 ) Suscessful for numerical methods for fractional equations with Riemman-Liouville and Pareto derivatives and for Grünwald-Letnikov, L1 and L2 discretization schemes, and many others Von Neumann procedure: 1: v (n) j = X q» q (n) e iqj x 2: Stability analysis of a generic subdiffusive mode:» (n) q Av (n) = Mv (n) h A Equation for the evolution of» (n) q :» (n) q F h e iqj xi h = M» fng q i ; S(q) = 0» (n) q e iqj x e iqj xi

28 Does the method always work? Stability First example: von-neumann stability analysis of the Yuste-Acedo method (SIAM J. Num. Anal. 05, fixed timesteps) for Riemann-Liouville FDE RL derivative S(q) 4 ( t) ( x) 2 sin2 µ q x 2 u(x; t) = K 0D 2 t u(x; (n+1) =» (n) S(q) nx k=0! (1 ) k» (n k) h i F» fn+1g ; S(q) = 0 (*1) Eq. of evolution of the amplitude of a generic mode. Similar to those of (non-fractional) multistep (multilevel) schemes. This equation and/or fractional difference methods could be seen as particular cases of multilevel schemes where the number of levels increases with time.

29 Does the method always work? Stability First example: Yuste-Acedo method (SIAM J. Num. Anal. 05, fixed timesteps) for RL FDE mx (*1)» (m+1) =» (m) S(q)! (1 )» (m k) F k=0 k h i» fng ; S(q) = 0 (*1) Stable if the n roots of Stability polynomial z n = z n 1 S(q) n 1 X k=0! (1 ) k z n 1 k satisfy jz n j 1 S(q) = S(q; n) z n + z n 1 P n 1 k=0!(1 ) k z n 1 k Stability region: region (in the complex plane ) formed by those values of S(q; n) for which jz n j 1 S(q) 4 ( t) ( x) 2 sin2 µ q x 2 Boundary of the stability region: z = e iµ S(q)! S B (q; µ) with µ = 0! µ = 2¼

30 Does the method always work? Stability First example: Yuste-Acedo method (SIAM J. Num. Anal. 05, fixed timesteps, explicit) for RL FDE S(q; n) z n+1 + z n iµ z = e P n k=0!(1 ) k z n k S B (q; µ; n) ei(n+1)µ + e inµ P n k=0!(1 ) k e i(n k)µ S B (q; µ; n) γ=1/2 S B (q; µ; 100) Stable 2 γ S(q) 4 ( t) ( x) 2 sin2 µ q x 2 Numerical method for RL-FDE: stable when 4 ( t) ( x) 2 S = 2 Yuste-Acedo arxiv:cs/ v1 [cs.na] (2003)

31 Does the method always work? Stability Second example: von-neumann stability analysis of the Liu-Zhuang-Anh-Turner method* (ANZIAM J. 47 (2006),fixed timesteps, implicit) for Caputo FDE *=present method for fixed timesteps S B (q; µ; = 2 Stability polynomial S(q; n) z n + z n 1 P n 1 k=1 [(k + 1)1 k 1 ] z n k S B (q; µ; 100) Estable Numerical method stable for all S(q)>0 µ unconditionally stable S(q) 4 ( t) q x ( x) 2 sin2 2

32 Does the method always work? Stability A zoo of stability regions RL FDE, weighted-average method, (Yuste, Journal of Computational Physics (2006)) Fully implicit S B (q; µ; 100) (fractional) Crank-Nicholson Explicit S B (q; µ; 100) Unconditionally stable Unconditionally stable conditionally stable

33 Does the method always work? Stability Does the (fractional) von-neumann stability analysis work for variable timesteps? Present method (Caputo FDE, implicit) with variable timesteps: t m = (m + 2)=10 S B (q; µ; 100) Unconditionally stable Fixed timesteps

34 Does the method always work? Stability Does the (fractional) von-neumann stability analysis work for variable timesteps? Present method (Caputo FDE, implicit) with random variable timesteps: t m = r=10 S B (q; µ; n) r=random number, uniform in [0,2] S B (q; µ; 10) S B (q; µ; 20) S B (q; µ; 30) S B (q; µ; 50) Unconditionally stable

35 Does the method always work? Stability The present implicit method for the FDE in the Caputo form with variable timesteps is unconditionally stable for any choice of timesteps (Yuste&Quintana-Murillo, Computer Physics Communications, Vol. 183, December 2012) It is proved there that v (n) 2 v (0) 2 always!

36 Adaptive finite difference method for Diffusion-wave = u + F(x; 1 < < 2 78

37 Finite difference method: Discretization of the ± ± t K± 2 x ± 2 xu(x j ; t) = u(x j+1; t) 2u(x j ; t) + u(x j 1 ; t) ( x) 2 Discretization of the Laplacian: three point centered formula 1<γ<2 : diffusion-wave ± t u(x; t n ) = 1 (3 ) n 1 X m=0 n A ( ) m;n [u(x; t m+1 ) u(x; t m )] o B m;n ( ) [u(x; t m ) u(x; t m 1 )] L2 discretization of the fractional Caputo derivative S n U n j+1 + (1 + 2S n )U n j S n U n j 1 = MU n j + ~ F(x j ; t n ) MU n j U (n 1) j + t n t h n 1 U (n 1) j t n 1 t n 2 n 2 X m=0 ³ ^Am;n h U (m+1) j i U (n 2) j i U (m) j ^B h m;n U (m) j i U (m 1) j S n = (3 )K(t n t n 2 ) 2(t n t n 1 ) 1 ( x) 2

38 The testbed problem u(x,0) u(x,t)=? = 2 0 x ¼ u(x = 0; t) = u(x = ¼; t) = 0 u(x; 0) = sin t)=@tj t=0 = 0 K = 1 u(x; t) = E ( t ) sin(x) 82

39 Step doubling algorithm Numerical results γ=3/2 Solid line: exact solution; symbols: adaptive method with = 10 4 up to time t 100 = 13:1958 (squares), = 10 5 up to time t 150 = 11:3309 (circles), and = 10 6 up to time t 300 = 9:9783 (triangles). In all cases x = ¼=40 and 0 = 0:01

40 Step doubling algorithm Numerical results γ=3/2 Numerical errors at the mid-point, ju(¼=2; t n ) Uk n j, of the adaptive method for = (squares), = 10 5 (circles), and = 10 6 (triangles). In all cases 0 = 0:01 and x = ¼=40.

41 Step doubling algorithm Numerical results γ=3/2 Exact solution (lines) and adaptive numerical solution (symbols) of the di usionwave problem described in the main text for = 3=2 and t 16 = 0:105 (squares), t 84 = 1:006 (circles), t 156 = 3:015 (stars), and t 233 = 5:741 (triangles), with = 10 6, 0 = 0:01 and x = ¼=40.

42 Numerical results. Diffusion-wave equation Some remarks Disappointing results (in comparison with those for subdiffusion equations) Results worsens when the lengths of two successive timesteps, n and n 1, are too di erent We cap the ratio between the lengths of two successive timesteps, [ j n = m 1 j or j n 1 = n j] to be smaller than 1:1. R=1.02 (really timid exploration coefficient ) Numerical tests Worsens? Is the method stable? We have carried out extensive calculations and considered a large variety of timestep functions (including random distributions), and we have always found well-behaved (stable) numerical solutions

43 Diffusion-wave equation. Boundaries of stability for some methods with fixed timesteps The stability analysis works fine but Explicit Gorenflo,Mainardi, Moretti & Paradisi 2002, Nonlinear Dyn.; also, Quintana-Murillo & Yuste, 2009, Physica Scripta Explicit Quintana-Murillo & Yuste.,2011. J. Comp.Nonlin. Dyn. Implicit Present method for fixed timesteps

44 Diffusion-wave equation. Boundaries of stability for variable timesteps S B (q; µ; n) n=50 An example t m = (m + 2)=100? However S(q) 4 ( t) ( x) 2 sin2 µ q x 2 well-behaved (stable) numerical solution t m = (m + 2)=100

45 Remarks and conclusions Computational cost of fractional difference methods huge Different time scales usual Adaptive finite difference method with non-uniform timesteps {very convenient a must} von Neumann stability analysis for homogeneous timesteps a breeze von Neumann stability analysis for variable timesteps: for subdifusion equations works smoothly for diffusion-wave equations? Adaptive step doubling method fast and accurate (for subdiffusion equations) 89