The convection diffusion equation¶
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
pl.style.use('seaborn-white')
%matplotlib inline
The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. (Wikipedia)
The equation reads
with
- \(U\) the physical quantities transferred (it could be a chemical species concentration, the temperature of a fluid...)
- \(k\) a diffusion convection
- \(c\) a velocity, which will be constant in our example.
model = Model("k * dxxU - c * dxU",
"U", ["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 three gaussian pulses for the initial condition
U = (np.roll(gaussian(x.size, 10), x.size // 5) +
np.roll(gaussian(x.size, 10), -x.size // 5) -
gaussian(x.size, 20))
fields = model.fields_template(x=x, U=U)
pl.figure(figsize=(15, 4))
pl.plot(fields.x, fields.U)
pl.xlim(0, fields.x.max())
pl.show()

We precise our parameters. The default scheme provide an automatic time_stepping. We set the periodic flag to True.
parameters = dict(k=.2, c=10, periodic=True)
We initialize the simulation.
t = 0
simulation = Simulation(model, t, fields, parameters,
dt=.1, tmax=30)
We iterate on the simulation until the end.
pl.figure(figsize=(15, 10))
for i, (t, fields) in enumerate(simulation):
if i % 2 == 0:
pl.fill_between(fields.x, fields.U + .1 * (i + 1),
fields.U.min() - 1,
color='darkred', zorder=-2 * i, alpha=.7)
pl.plot(fields.x, fields.U + .1 * (i + 1),
color='white',
zorder=-(2 * i - 1))
print(f"t: {t:g}".ljust(80), end='\r')
pl.xlim(0, fields.x.max())
pl.show()
t: 29.9
