Simulation and visualization

of matrix hydrodynamics

 

github.com/klasmodin/quflow

Installation

git clone https://github.com/klasmodin/quflow.git
cd quflow
pip install .

With conda (recommended)

download the quflow conda environment file

then run the following command:

conda env create -f quflow-env.yaml
# Below only on Apple Silicon hardware:
conda install "libblas=*=*accelerate"

With pip

Basic concepts

Classical

function \(\omega:S^2\to\mathbb{R}\)

spherical harmonics \(\omega = \sum_{l=0}^\infty\sum_{m=-l}^l\omega^{lm}Y_{lm}\)

Quantized

complex matrix \(W\in\mathfrak{u}(N)\)

matrix harmonics \(W = \sum_{l=0}^N\sum_{m=-l}^l\omega^{lm}T_{lm}\)

import quflow as qf
omega = np.random.randn(10)
N = 512
W = qf.shr2mat(omega, N)
qf.plot(W)

real spherical harmonics

Basic concepts, continued

vorticity form of 2-D Euler

Classical Matrix correspondence
Poisson bracket Scaled commutator
Poisson equation Matrix equation

\(\{\omega,\psi\}\)

\(\frac{1}{\hbar}[W,P]\)

\(\Delta\psi = \omega \)

\(\Delta_N P = W \)

\dot\omega + \{\psi,\omega\} = 0, \quad\Delta\psi = \omega

becomes

\dot W + \frac{1}{\hbar}[P,W] = 0, \quad\Delta_N P = W
\frac{2}{\sqrt{N^2-1}}

Hoppe-Yau Laplacian

Pre-processing

  • create initial state \(W_0\in\mathfrak{su}(N)\)
  • specify data filename (HDF5)
  • create QuSimulation object
  • specify parameters
import quflow as qf
N = 128
W0 = qf.shr2mat(np.random.randn(10), N)
filename = "basic_euler_N_{}.hdf5".format(N)

# Create simulation object associated with `filename`
mysim = qf.QuSimulation(filename, overwrite=True, state=W0, 
                        loggers={'energy':qf.energy_euler, 
                                 'enstrophy':qf.enstrophy})
# Set parameters
mysim['dt'] = 0.5*qf.hbar(N)  # scale by hbar to get independence of N
mysim['simtime'] = 5.0
mysim['inner_time'] = 0.2  # how often to write output
mysim['hamiltonian'] = qf.solve_poisson  # which Hamiltonian to use

Running simulation

  • move data file to cluster (optional)
  • adjust total simulation time (optional)
  • run the simulation
import quflow as qf
N = 128
filename = "basic_euler_N_{}.hdf5".format(N)

# Read simulation object associated with `filename`
mysim = qf.QuSimulation(filename)

# Adjust simulation time if needed
mysim['simtime'] = 10.0

# Run the simulation
qf.solve(mysim)

(repeat the above to continue simulation longer)

Post-processing

  • move data file from cluster (optional)
  • plot and/or animate the data
import quflow as qf
N = 128
filename = "basic_euler_N_{}.hdf5".format(N)

# Read simulation object associated with `filename`
mysim = qf.QuSimulation(filename)

# Plot the last state 
# ('fun' is the output of function, 'mat' is the pure matrix)
qf.plot(mysim['fun', -1])

# Create animation of the entire simulation
anim = qf.create_animation(filename.replace(".hdf5",".mp4"), mysim['fun', :])
anim

Result for generic

smooth initial data

Results for generic vanishing momentum smooth initial data

Brief demo of QUFLOW

By Klas Modin

Brief demo of QUFLOW

Demonstration of the Python software QUFLOW for simulation of Euler's equations on the sphere.

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