Coupled burger’s-like equations

import functools as ft
import multiprocessing as mp
import logging

import numpy as np
from scipy.signal import gaussian

import pylab as pl

from triflow import Model, Simulation, schemes, displays

%matplotlib inline

We initialize the model with a coupled burger-like system of equations

model = Model(["k * dxxU - c * U * dxV", "k * dxxV - c * V * dxU"],
               ["U", "V"], ["k", "c"])

We discretize our spatial domain. retstep=True ask to return the spatial step. We want periodic condition, so endpoint=True exclude the final node (which will be redondant with the first node, \(x=0\) and \(x=100\) are merged)

x, dx = np.linspace(0, 100, 500, retstep=True, endpoint=False)

We initialize with cosine and sine function for \(U\) and \(V\).

U = np.cos(x * 2 * np.pi / x.max() * 5) * .5 + 1
V = np.sin(x * 2 * np.pi / x.max() * 5) * .5 + 1
fields = model.fields_template(x=x, U=U, V=V)

We precise our parameters. The default scheme provide an automatic time_stepping. We set the periodic flag to True.

parameters = dict(k=1, c=10, periodic=True)

We initialize the simulation

t = 0
simulation = Simulation(model, t, fields, parameters,
                        dt=.1, tmax=4, tol=1E-2)

We iterate on the simulation until the end.

pl.figure(figsize=(20, 12))
for i, (t, fields) in enumerate(simulation):
    if i % 2 == 0:
        pl.fill_between(fields.x, fields.U + .1 * (2 * i),
                        fields.U.min() - 1,
                        color='darkred', zorder=-4 * i, alpha=.7)
        pl.plot(fields.x, fields.U + .1 * (2 * i),
                color='white',
                zorder=-4 * i + 1)
        pl.fill_between(fields.x, fields.V + .1 * (2 * i + 1),
                        fields.V.min() - 1,
                        color='lightsteelblue', zorder=-4 * i + 2, alpha=.7)
        pl.plot(fields.x, fields.V + .1 * (2 * i + 1),
                color='white',
                zorder=-4 * i + 3)
    print(f"t: {t:g}".ljust(80), end='\r')
pl.xlim(0, fields.x.max())
pl.show()
t: 3.9
../_images/so_wavy_13_1.png