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from IPython.display import HTML, display, Image
import numpy as np
import pylab as pl
import triflow as trf
from scipy.signal import gaussian
%matplotlib inline
This notebook is an attempt to reproduce the Dedalus Project tutorial. The equation to solve is
$$\partial_{t}\,U + U \partial_{x}\,U = a\partial_{xx}\,U + b\partial_{xxx}\,U$$
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model = trf.Model("-U * dxU + a * dxxU + b * dxxxU", "U", ["a", "b"])
We discretize our spatial domain. retstep=True ask to return the spatial step.
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x, dx = np.linspace(-2, 6, 5000, retstep=True, endpoint=False)
We initialize with the same initial condition as in the Dedalus tutorial
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n = 20
U = np.log(1 + np.cosh(n)**2/np.cosh(n*x)**2) / (2*n)
fields = model.fields_template(x=x, U=U)
pl.figure(figsize=(15, 4))
pl.plot(fields.x, fields.U)
pl.xlim(x.min(), fields.x.max())
pl.show()
We precise our parameters. The default scheme provide an automatic time_stepping.
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parameters = dict(a=2E-4, b=1E-4, periodic=False)
We initialize the simulation.
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%%opts Curve [show_grid=True, width=800] {+framewise}
t = 0
simulation = trf.Simulation(model, fields, parameters,
dt=.05, tmax=10, tol=1E-1)
container = simulation.attach_container()
trf.display_fields(simulation)
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We iterate on the simulation until the end.
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simulation.run()
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display(HTML("<h3>Triflow result</h3>"))
pl.figure(figsize=(12, 8))
container.data.U.plot(vmin=-.65, vmax=.95, cmap="RdBu_r")
pl.show()
display(HTML("<h3>Dedalus result</h3>"))
display(Image('dedalus-kdv.png'))
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