An overview of the BECASIM project: opensource numerical simulators for the

Gross-Pitaevskii equation

Ionut Danaila

Laboratoire de mathematiques Raphael SalemUniversite de Rouen, France

`mrs.univ-rouen.fr/Persopage/Danaila

New challenges in mathematical modelling and numericalsimulation of superfluids, CIRM, June 27, 2016.

Framework

ANR project BECASIM:BEC Advanced SIMulations

Agence Nationale de la Recherche

ANR Project BECASIM (Numerical Methods, 2013-2017)25 French mathematicians from 10 different labs

develop new methods for real and imaginary time GP,mathematical theory, numerical analysis,(HPC) parallel codes:: open source,huge simulations of physical configurations(turbulence in BEC).

becasim.math.cnrs.fr

Framework

Outline

1 ANR project BECASIMBECASIM teamMathematical modelsProgram of the BECASIM sessionNumerical tools

Framework

BECASIM team

BECASIM team: 9 labs, 25 researchers

• complementary skills: theory of PDEs /numericalanalysis/algorithms/ HPC, etc.

LilleI. Lacroix-

Violet

Rouen-ParisI. Danaila

ToulouseC. Besse

Nancy-MetzX. Antoine

MontpellierR. Carles

4 EC1 IR

3 EC

2 EC1 IR

4 EC1 IR

4 EC1 IR

Framework

Mathematical models

Gross-Pitaevskii (GP) equation(s)

Unsteady GP→ real time dynamics

i~∂ψ

∂t= − ~2

2m∇2ψ + Ng3D|ψ|2ψ + Vtrap ψ

mean field theory : ψ order parameter,

nonlinear Schrodinger equation (cubic nonl, defocusing),

conservation laws: number of atoms∫|ψ|2 and energy E(ψ).

Steady GP→ ground and meta-stable statesψ = φexp(−iµt/~), µ is the chemical potential

− ~2

2m∇2φ+ Vtrap φ+ Ng3D|φ|2φ− µφ = 0

nonlinear eigenvalue problem,

conservation laws: number of atoms∫|ψ|2.

But also (see becasim.math.cnrs.fr)two-component BEC, non-local potentials,stochastic GP, fractional GP, etc

Framework

Mathematical models

Gross-Pitaevskii (GP) equation(s)

Unsteady GP→ real time dynamics

i~∂ψ

∂t= − ~2

2m∇2ψ + Ng3D|ψ|2ψ + Vtrap ψ

mean field theory : ψ order parameter,

nonlinear Schrodinger equation (cubic nonl, defocusing),

conservation laws: number of atoms∫|ψ|2 and energy E(ψ).

Steady GP→ ground and meta-stable statesψ = φexp(−iµt/~), µ is the chemical potential

− ~2

2m∇2φ+ Vtrap φ+ Ng3D|φ|2φ− µφ = 0

nonlinear eigenvalue problem,

conservation laws: number of atoms∫|ψ|2.

But also (see becasim.math.cnrs.fr)two-component BEC, non-local potentials,stochastic GP, fractional GP, etc

Framework

Program of the BECASIM session

Program of the BECASIM session

1 X. Antoine: GPELab, an open source Matlab toolbox for thenumerical simulation of Gross-Pitaevskii equations

2 Q. Tang: Numerical methods on simulating dynamics of thenonlinear Schrodinger equation with rotation and/or nonlocalinteractions

3 C. Besse: High-order numerical schemes for computing thedynamics of nonlinear Schrodinger equation

4 P. Parnaudeau: A hybrid code for solving the Gross-Pitaevskiiequation

5 R. Carles: Time splitting methods and the semi-classical limit6 A. de Bouard: Inhomogeneities and temperature effects in

Bose-Einstein condensates

Framework

Numerical tools

Matlab toolbox: GPELabGPELab (Fourier spectral, FFT)Developers : R. Duboscq, X. Antoine.• stationary GP: semi-implicit Euler,• real-time GP: splitting, relaxation,• stochastic GP: splitting, relaxation.Great flexibility to deal with new physical models:multi-component BEC BEC with double-well potential

Framework

Numerical tools

FreeFem++ Toolbox (www.freefem.org)Developers: G. Vergez, I. Danaila, F. Hecht.paper in revision, CCP (to freely distribute scripts)!

GPFEM: finite element solver2D/3D anisotropic mesh adaptation, flexibility for boundary conditions,• stationary GP: different Sobolev gradients.• instationary GP: splitting, relaxation schemes.

Framework

Numerical tools

FreeFem++: Bogoliubov-de Gennes modes

Two-component condensate:

i~∂ψ1

∂t=

[− ~2

2m∇2 + Vtrap(x) + g11|ψ1|2 + g12|ψ2|2

]ψ1,

i~∂ψ2

∂t=

[− ~2

2m∇2 + Vtrap(x) + g21|ψ1|2 + g22|ψ2|2

]ψ2.

The Bogoliubov-de Gennes model is based on the linearisation:

ψ1(x, t) = exp(−iµ1t/~)(φ1 + a(x)e−iωt + b∗(x)eiω∗t

)ψ2(x, t) = exp(−iµ2t/~)

(φ2 + c(x)e−iωt + d∗(x)eiω∗t

)

Framework

Numerical tools

BdG 2d: Vortex-Antidark Solitary Waves

I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels, P. G.Kevrekidis, http://arxiv.org/abs/1606.05607

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0.5

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0.9

g12

/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Framework

Numerical tools

BdG 2d: Ring-Antidark-Ring Solitary Waves

I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels, P. G.Kevrekidis, http://arxiv.org/abs/1606.05607

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0.5

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0.9

g12

Re

al(

/)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

g12

Ima

g(

/)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

Framework

Numerical tools

BdG 2d: mesh adaptivity

I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels, P. G.Kevrekidis, http://arxiv.org/abs/1606.05607

Vortex-Antidark

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0.5

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0.9

x

y

­15 ­10 ­5 0 5 10 15­15

­10

­5

0

5

10

15g=0.5

x

y

­15 ­10 ­5 0 5 10 15­15

­10

­5

0

5

10

15g=0.9

Ring-Antidark-Ring

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0.5

x

|1|2

, |

2|2

­15 ­10 ­5 0 5 10 15

0

0.5

1

1.5

2g

12=0.9

x

y

­15 ­10 ­5 0 5 10 15­15

­10

­5

0

5

10

15g=0.5

x

y

­15 ­10 ­5 0 5 10 15­15

­10

­5

0

5

10

15g=0.9

Framework

Numerical tools

MPI-OpenMP code (GPS): 6th order FD or spectral

Developers: Ph. Parnaudeau, A. Suzuki, J.-M Sac-Epee.

• stationary GP: backward semi-implicit Euler, Sobolev gradients.• real-time GP: splitting, relaxation, Crank-Nicolson.Flexible to run on laptops→ clusters: 2D/3D grids up to 20483,optimized for OpenMP-MPI, from 4→ 105 cores.

0.1

1

10

100

100 1000 10000 100000

spee

dup

#cores

5123

10243

20483

Framework

Numerical tools

2005 3D Simulation: grid 2403, 2 weeks)

I. Danaila, Phys. Rev. A, 2005.

Framework

Numerical tools

2014 3D Simulation: grid 5123, 1 day

Ph. Parnaudeau, CPC, to be submitted.

Framework

Numerical tools

Conclusion

Project ANR BECASIM (Numerical Methods, 2013-2016)25 French mathematicians from 10 different labs

becasim.math.cnrs.fr

Messages to physicistsWe develop new numerical methods and HPC codes:

currently under intensive tests,will be distributed as open source,we seek challenging physical applications.