Automatic finite difference discretization for 1D PDE with fast temporal solvers.
Project description
Scikit-fdiff / skfdiff
The full documentation is available on read the doc.
Installation
External requirements
This library is written for python >= 3.7.
On v0.7.0, it is possible to choose between numpy and numba (which provide similar features). numpy will be slower but with no compilation time, which is handy for testing and prototyping. On other hand, numba use a JIT compilation, and give access to faster and parallized routines in the cost of an extra dependency and a warm-up time.
via PyPI
pip install scikit-fdiff[numba,interactive]
will install the package and
pip install scikit-fdiff --upgrade
will update an old version of the library.
via github
You can install the latest version of the library using pip and the github repository:
pip install git+https://gitlab.com/celliern/scikit-fdiff
Introduction
Rational
The aim of this library is to have a (relatively) easy way to write transient systems of N-dimensional partial differential equations with finite difference discretization and fast temporal solvers.
The main two parts of the library are:
- symbolic tools defining the spatial discretization, with boundary taking into account in a separated part
- a fast temporal solver written in order to use the sparsity of the finite difference method to reduce the memory and CPU usage during the solving
Moreover, extra tools are provided and the library is written in a modular way, allowing an easy extension of these different parts (see the plug-in module of the library.)
The library fits well with an interactive usage (in a jupyter notebook). The dependency list is actually larger, but on-going work target a reduction of the stack complexity.
Model writing
All the models are written as function generating the F
vector and the Jacobian matrix of the model defined as dtU = F(U)
.
The symbolic model is written as a simple mathematic equation. For example, a diffusion advection model can be written as:
from skfdiff import Model
equation_diff = "k * dxxU - c * dxU"
dependent_var = "U"
physical_parameters = ["k", "c"]
model = Model(equation_diff, dependent_var,
physical_parameters)
Toy examples (more ambitious one are in the doc)
1D advection / diffusion system, Dirichlet boundary
>>> import pylab as pl
>>> import numpy as np
>>> from skfdiff import Model, Simulation
>>> model = Model("k * dxxU - c * dxU",
... "U(x)", ["k", "c"],
... boundary_conditions={("U", "x"): ("dirichlet", "dirichlet")}
... )
>>> x, dx = np.linspace(0, 1, 200, retstep=True)
>>> U = np.cos(2 * np.pi * x * 5)
# The fields are ``xarray.Dataset`` objects, and all the
# tools / methods available in the ``xarray`` lib can be
# applied to the skfdiff.Fields.
>>> fields = model.Fields(x=x, U=U, k=0.001, c=0.3)
# fix the boundary values for the dirichlet condition
>>> fields["U"][0] = 1
>>> fields["U"][-1] = 0
>>> t = 0
>>> dt = 5E-1
>>> tmax = 2.5
>>> simul = Simulation(model, fields, dt, tmax=tmax)
# The containers are in-memory or persistant
# xarray Dataset with all or some time-steps available.
>>> container = simul.attach_container()
>>> simul.run()
(2.5, <xarray.Dataset>
Dimensions: (x: 200)
Coordinates:
* x (x) float64 0.0 ... 1.0
Data variables:
U (x) float64 ...
k float64 0.001
c float64 0.3)
>>> for t in container.data.t:
... fig = pl.figure()
... plot = container.data["U"].sel(t=t).plot()
2D advection / diffusion system, mixed robin / periodic boundary
>>> import pylab as pl
>>> import numpy as np
>>> from skfdiff import Model, Simulation
# some specialized scheme as the upwind scheme as been implemented.
# as the problem as a strong advective component, we can use it
# to reduce the numerical instabilities.
# the dirichlet condition mean that the boundary will stay fix,
# keeping the initial value.
>>> model = Model("k * (dxxU + dyyU) - (upwind(cx, U, x, 2) + upwind(cy, U, y, 2))",
... "U(x, y)", ["k", "cx", "cy"],
... boundary_conditions={("U", "x"): ("dxU - (U - sin(y) * cos(t))", "dxU - 5"),
... ("U", "y"): "periodic"})
>>> x = np.linspace(0, 10, 56)
>>> y = np.linspace(-np.pi, np.pi, 32)
>>> U = np.zeros((x.size, y.size))
>>> fields = model.Fields(x=x, y=y, U=U, k=0.001, cx=0.8, cy=0.3)
>>> dt = 1.
>>> tmax = 15.
>>> simul = Simulation(model, fields, dt, tmax=tmax, tol=5E-1)
>>> container = simul.attach_container()
>>> simul.run()
(15.0, <xarray.Dataset>
Dimensions: (x: 56, y: 32)
Coordinates:
* x (x) float64 0.0 ... 10.0
* y (y) float64 -3.142 ... 3.142
Data variables:
U (x, y) float64 ...
k float64 0.001
cx float64 0.8
cy float64 0.3)
>>> for t in np.linspace(0, tmax, 5):
... fig = pl.figure()
... plot = container.data["U"].sel(t=t, method="nearest").plot()
Contributing
See the contribute section of the doc.
Code of Conduct
See the dedicated page.
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