Automatic finite difference discretization for 1D PDE with fast temporal solvers.

Scikit-fdiff / skfdiff (formerly Triflow)

The full documentation is available on read the doc.

master dev

Installation

External requirements

This library is written for python >= 3.6.

On v0.6.0, 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 skfdiff[numba,interactive]


will install the package and

pip install skfdiff --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+git://github.com/locie/skfdiff.git


Introduction

Rational

The aim of this library is to have a (relatively) easy way to write transient dynamic systems finite difference discretization, with 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()


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