Numerical simulation of the Gross-Pitaevskii equation by pseudo-spectral and finite element methods comparison of GPS code and FreeFem++

Transcription

1 . p.1/13 10 ec. 2014, LJLL, Paris FreeFem++ workshop : BECASIM session Numerical simulation of the Gross-Pitaevskii equation by pseudo-spectral and finite element methods comparison of GPS code and FreeFem++ Atsushi Suzuki LJLL, INRIA-Nancy

2 stationary state of Gross-Pitaevskii equation 1/3 time-dependent Gross-Pitaevskii equation in 2, R 2 i t φ(t) = 1 2 φ(t) Ωi` y φ(t) + V φ(t) + β φ(t) 2 φ(t) in (0, T) φ(0, ) = φ 0 stationary state is given by the minimization of the energy φ H0(, 1 C) 1 E(φ) = 2 φ 2 + V φ 2 + β 2 φ 4 y Ωi` φ φ min E(φ). φ =1 Remark on energy of the angular momentum term R ` y φ φ R (y x φ x y φ) φ = ( x (y φ) y (x φ))φ + = {x ; x < R} `n x n y = `x y /R on y nx x n y = 0. Re (y x φ x y φ) φ = 0 (y n x x n y )φ φ. p.2/13

3 . p.3/13 stationary state of Gross-Pitaevskii equation 2/3 minimization problem E(φ) = min E(φ). φ =1 equivalent nonlinear eigen value problem 1 2 φ 2 + V φ 2 + β 2 φ 4 Ωi` y φ φ find λ R, φ H0 1 ( ; C) λφ = 1 2 φ Ωi` y φ + V φ + β φ 2 φ φ = 1 λ : chemical potential λ = λφ φ = =E(φ) + β 2 φ 4 φ φ Ωi` y φ φ + V φ φ + β φ 2 φ φ

4 . p.4/13 stationary state of Gross-Pitaevskii equation 3/3 minimization problem E(φ) = min E(φ). φ =1 1 2 φ 2 + V φ 2 + β 2 φ 4 Ωi` y φ φ normalized gradient flow ( imaginary time method : i t t ) φ t = 1 2 E(φ) φ φ(t n+1, ) = φ(t n+1, ) φ(t n+1, ) φ(0, ) = φ 0, φ 0 = 1 = φ + Ωi` y φ V φ β φ 2 φ t [t n, t n+1 ) potential α = 1.2, κ = 0.3 V (x, y) = 1 α 2 (x 2 + y 2 ) + κ 4 (x2 + y 2 ) 2 appropriate initial condition : Thomas-Fermi approximation

5 . p.5/13 discretization scheme semi-implicit Euler scheme for time discretization φ n+1 φ n t φ n+1 = = φ n+1 y + Ωi` x φn+1 V φ n+1 β φ n 2 φn+1 φ n+1 φ n+1 φ 0 = φ 0, φ 0 = 1 Space discretization pseudo-spectral method expansion by Fourier basis + collocation method finite element method P1 element + mesh adaptation by FreeFem++

6 . p.6/13 pseudo-spectral method 1/2 = ( π, π) ( π, π) and assume φ is periodic in x- and y-directions expansion by Fourier basis φ(x, y) P ˆφ N/2 k<n/2 k (y)e i kx ˆφ k (y) = 1 2π 1 2π R π π 2π N φ(x, y)e i kx P 0 p<n φ(x p, y)e i kx p, x p = π + 2π N p xφ(x, 2 y) = x 2 Pk ˆφ k (y)e i kx = P k ( k2 )ˆφ k (y)e i kx y x φ(x, y) = y x Pk ˆφ k (y)e i kx = y P k i kˆφ k (y)e i kx at collocation points (x r, y s ) ( same as integration points ) 1 t 1 2 Ωi` y r x s + V (xr, y s ) + β φ n r,s 2 acts to φ n+1 r,s and it is evaluated using xφ(x, 2 y) (xr,y s ) = P k ( k2 ) 1 P N p φ p,se i kx pe i kx r y x φ(x, y) (xr,y s ) = y s Pk i k 1 P N p φ p,se i kx pe i kx r these computations are realized by 1 FFT (fftw library)

7 . p.7/13 pseudo-spectral method 2/2 The operator on collocation points (x r, y s ) ( 1 t 1 2 Ωi` y r x + V (xr, y s s ) + β φ n r,s 2 )φ n+1 r,s collocation method for angular momentum, potential and nonlinear interaction terms simplifies the code there is no explicit matrix expression of the linear operator. The matrix is dense because of forward and backward FFTs. Linear solver Krylov subspace method with Thomas-Fermi preconditioner (X. Antoine, R. uboscq) (P r,s ) 1 = ( 1 t + V (x r, y s ) + β φ n r,s 2 ) 1 GPS code (P. Parnaudeau, J-M. Sac-Epée, A. Suzuki) BiCGStab, GCR solver written by Fortran90 using fftw and OpenMP + MPI (for 3) cut-off with some radius to satisfy homogeneous irichlet boundary conditions parallelization technique is encapsulated in vector-matrix product and inner product

8 finite element method real dt=0.005, alpha=1.2,kappa=0.3; func V=(1.0-alpha)/2.0*(x*x+y*y)+kappa/4.0*(x*x+y*y)*(x*x+y*y); real Omega=3.5, beta=1000.0; func Vtrap=(x*x+y*y)*0.5; real mutf=sqrt(beta/pi); func rhotf=max(0.0,(mutf-vtrap)/beta); fespace Vh(Th,P1); Vh<complex> u,v,u0; Vh zz; u0=sqrt(rhotf); varf gpe(u,v)=int2d(th)(1.0/dt*u*conj(v) +0.5*(dx(u)*conj(dx(v))+dy(u)*conj(dy(v))) -Omega*1i*(y*dx(u)-x*dy(u))*conj(v) +(V+beta*u0*conj(u0))*u*conj(v))+on(1,u=0.0); varf rhs(u,v)=int2d(th)(1.0/dt*u0*conj(v))+on(1,u=0.0); for (int it=0; it < 10000; ++it) { matrix<complex> A=gpe(Vh,Vh,tgv=-1); set(a,solver=sparsesolver,tgv=-1); complex[int] b=rhs(0,vh,tgv=-1); complex[int] w=aˆ-1*b; zz=real(u0)*real(u0)+imag(u0)*imag(u0); u0=u0/sqrt(int2d(th)(zz)); }. p.8/13

9 mesh adaptation after some time steps. p.9/13

10 . p.10/13 Comparison of performance FreeFem++, UMFPACK t = 0.005, n = 274, matrix build RHS build factorization fw/bw subst. energy comput. # CPU FreeFem++, dissection t = 0.005, n = 274, GPS/fftw, t = 0.005, n = 512 2, #itr GCR= / / GPS/fftw, t = , n = 512 2, #itr GCR= / / mac mini 2012, Intel Core i7 2.3GHz + 16G mem. Intel ifort

11 . p.11/13 Newton-Raphson method for nonlinear eigen value problem 1/3 Is stationary state obtained by gradient flow? dependence of stationary state on the initial condition seems to be not clear. How to understand stationary state? Newton-Raphson nonlinear solver approach F 1 (φ, λ) := λφ φ + Ω i` y φ V φ β φ 2 φ F 2 (φ, λ) := λ( φ 2 1) `φ0 0 λ : given by gradient flow solution, with computing λ 0 as chemical potential loop n = 0, 1,2, find `δ ε by solving a linear system: (λ n Ω i` y V 2β φ n 2 )δ β(φ n ) 2 δ + εφ n = F 1 (φ n, λ n ) λ n ( φ n δ + φ n δ) + ε( φ n 2 1) = F 2 (φ n, λ n ) `φn+1 n+1 λ = `φn n λ `δ ε

12 . p.12/13 Newton-Raphson method for nonlinear eigen value problem 2/3 Linear system with (N + 1) (N + 1) matrix (λ n Ω i` y V 2β φ n 2 )δ β(φ n ) 2 δ + εφ n = F 1 (φ n, λ n ) λ n ( φ n δ + φ n δ) + ε( φ n 2 1) = F 2 (φ n, λ n ) block factorization strategy using N N linear solver: solve two linear systems (λ n Ω i` y x V 2β φ n 2 )σ β(φ n ) 2 σ = φ n (λ n Ω i` y V 2β φ n 2 )γ β(φ n ) 2 γ = F 1 (φ n, λ n ) solve Schur complement problem sε = λ n ( φ n 2 1) λ n backward substitution φ n γ + φ n γ (λ n Ω i` y x V 2β φ n 2 )γ β(φ n ) 2 γ = F 1 (φ n, λ n ) εφ n

13 . p.13/13 Newton-Raphson method for nonlinear eigen value problem 3/ l2 norm of updating iteration E(φ) λ # vortex GPS FreeFem