# Source code for gemseo.problems.dataset.burgers

```
# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
# Contributors:
# INITIAL AUTHORS - initial API and implementation and/or initial
# documentation
# :author: Syver Doving Agdestein
# OTHER AUTHORS - MACROSCOPIC CHANGES
r"""Burgers dataset.
This :class:`.Dataset` contains solutions to the Burgers' equation with
periodic boundary conditions on the interval :math:`[0, 2\pi]` for different
time steps:
.. math::
u_t + u u_x = \nu u_{xx},
An analytical expression can be obtained for the solution, using the Cole-Hopf
transform:
.. math::
u(t, x) = - 2 \nu \frac{\phi'}{\phi},
where :math:`\phi` is solution to the heat equation
:math:`\phi_t = \nu \phi_{xx}`.
This :class:`.Dataset` is based on a full-factorial
design of experiments. Each sample corresponds to a given time step :math:`t`,
while each feature corresponds to a given spatial point :math:`x`.
`More information about Burgers' equation
<https://en.wikipedia.org/wiki/Burgers%27_equation>`_
"""
from __future__ import annotations
from numpy import exp
from numpy import hstack
from numpy import linspace
from numpy import newaxis
from numpy import pi
from numpy import square
from gemseo.datasets.io_dataset import IODataset
[docs]
def create_burgers_dataset(
n_samples: int = 30,
n_x: int = 501,
fluid_viscosity: float = 0.1,
categorize: bool = True,
) -> IODataset:
"""Burgers dataset parametrization.
Args:
n_samples: The number of samples.
n_x: The number of spatial points.
fluid_viscosity: The fluid viscosity.
categorize: Whether to distinguish
between the different groups of variables.
Returns:
The Burgers dataset.
"""
time = linspace(0, 2, n_samples)[:, newaxis]
space = linspace(0, 2 * pi, n_x)[newaxis, :]
visc = fluid_viscosity
alpha = space - 4 * time
alpha_2 = square(alpha)
beta = 4 * visc * (time + 1)
gamma = space - 4 * time - 2 * pi
gamma_2 = square(gamma)
phi = exp(-alpha_2 / beta) + exp(-gamma_2 / beta)
phi_deriv = -2 * alpha / beta * exp(-alpha_2 / beta)
phi_deriv -= 2 * gamma / beta * exp(-gamma_2 / beta)
u_t = -2 * visc / phi * phi_deriv
if categorize:
groups = {"t": IODataset.INPUT_GROUP, "u_t": IODataset.OUTPUT_GROUP}
else:
groups = None
data = hstack([time, u_t])
dataset = IODataset.from_array(data, ["t", "u_t"], {"t": 1, "u_t": n_x}, groups)
dataset.name = "Burgers"
dataset.misc["x"] = [[node] for node in space[0]]
dataset.misc["nu"] = visc
return dataset
```