Simulation and visualization

of spherical hydrodynamics

 

github.com/klasmodin/quflow

Installation

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

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)

Units of time

Classical time: seconds

Quantum time: qtime

1 second = \(\displaystyle\frac{1}{\sqrt{16\pi}}N^{3/2}\) qtime

General rule:

  • numerical parameters use qtime (e.g. stepsize)
  • physical parameters use second (e.g. total simulation time)
import quflow as qf
N = 512
print(qf.seconds2qtime(1.0, N))
print(qf.qtime2seconds(1.0, N))

Running simulations

High-level solve function:

  • initial data \(W_0\in\mathfrak{u}(N)\)
  • simulation step-size (qtime)
  • total simulation time (seconds)
  • inner simulation time (seconds)
    (how often callback)
  • callback function (typically QuData object)
import quflow as qf
N = 128
W0 = qf.shr2mat(np.random.randn(10), N)
h = 0.1
mysim = qf.QuData("mysim.hdf5")
mysim(W0, 0.0) # Save first output
qf.solve(W0, qstepsize=h, time=3.0, inner_time=0.2, callback=mysim)
mysim.flush()

Generic smooth initial data

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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