findiff

A Python package for finite difference numerical derivatives and partial differential equations in any number of dimensions.

Main Features

• Differentiate arrays of any number of dimensions along any axis with any desired accuracy order
• Accurate treatment of grid boundary
• Can handle arbitrary linear combinations of derivatives with constant and variable coefficients
• Fully vectorized for speed
• Matrix representations of arbitrary linear differential operators
• Solve partial differential equations with Dirichlet or Neumann boundary conditions
• Symbolic representation of finite difference schemes
• New in version 0.11: More comfortable API (keeping the old API available)
• New in version 0.12: Periodic boundary conditions for differential operators and PDEs.

Installation

pip install --upgrade findiff

Documentation and Examples

You can find the documentation of the code including examples of application at https://findiff.readthedocs.io/en/stable/.

Taking Derivatives

findiff allows to easily define derivative operators that you can apply to numpy arrays of any dimension.

Consider the simple 1D case of a equidistant grid with a first derivative $\displaystyle \frac{\partial}{\partial x}$ along the only axis (0):

import numpy as np
from findiff import Diff

# define the grid:
x = np.linspace(0, 1, 100)

# the array to differentiate:
f = np.sin(x)  # as an example

# Define the derivative:
d_dx = Diff(0, x[1] - x[0])

# Apply it:
df_dx = d_dx(f) 

Similarly, you can define partial derivatives along other axes, for example, if $z$ is the 2-axis, we can write $\frac{\partial}{\partial z}$ as:

Diff(2, dz)

Diff always creates a first derivative. For higher derivatives, you simply exponentiate them, for example for $\frac{\partial^2}{\partial_x^2}$

d2_dx2 = Diff(0, dx)**2

and apply it as before.

You can also define more general differential operators intuitively, like

$$ 2x \frac{\partial^3}{\partial x^2 \partial z} + 3 \sin(y)z^2 \frac{\partial^3}{\partial x \partial y^2} $$

which can be written as

# define the operator
diff_op = 2 * X * Diff(0)**2 * Diff(2) + 3 * sin(Y) * Z**2 * Diff(0) * Diff(1)**2

# set the grid you use (equidistant here)
diff_op.set_grid({0: dx, 1: dy, 2: dz})

# apply the operator
result = diff_op(f)

where X, Y, Z are numpy arrays with meshed grid points. Here you see that you can also define your grid lazily.

Of course, standard operators from vector calculus like gradient, divergence and curl are also available as shortcuts.

If one or more axis of your grid are periodic, you can specify that when defining the derivative or later when setting the grid. For example:

d_dx = Diff(0, dx, periodic=True)

# or later
d_dx = Diff(0)
d_dx.set_grid({0: {"h": dx, "periodic": True}})

More examples can be found here and in this blog.

Accuracy Control

When constructing an instance of Diff, you can request the desired accuracy order by setting the keyword argument acc. For example:

d_dx = Diff(0, dy, acc=4)
df_dx = d2_dx2(f)

Alternatively, you can also split operator definition and configuration:

d_dx = Diff(0, dx)
d_dx.set_accuracy(2)
df_dx = d2_dx2(f)

which comes in handy if you have a complicated expression of differential operators, because then you can specify it on the whole expression and it will be passed down to all basic operators.

If not specified, second order accuracy will be taken by default.

Finite Difference Coefficients

Sometimes you may want to have the raw finite difference coefficients. These can be obtained for any derivative and accuracy order using findiff.coefficients(deriv, acc). For instance,

import findiff
coefs = findiff.coefficients(deriv=3, acc=4, symbolic=True)

gives

{'backward': {'coefficients': [15/8, -13, 307/8, -62, 461/8, -29, 49/8],
              'offsets': [-6, -5, -4, -3, -2, -1, 0]},
 'center': {'coefficients': [1/8, -1, 13/8, 0, -13/8, 1, -1/8],
            'offsets': [-3, -2, -1, 0, 1, 2, 3]},
 'forward': {'coefficients': [-49/8, 29, -461/8, 62, -307/8, 13, -15/8],
             'offsets': [0, 1, 2, 3, 4, 5, 6]}}

If you want to specify the detailed offsets instead of the accuracy order, you can do this by setting the offset keyword argument:

import findiff
coefs = findiff.coefficients(deriv=2, offsets=[-2, 1, 0, 2, 3, 4, 7], symbolic=True)

The resulting accuracy order is computed and part of the output:

{'coefficients': [187/1620, -122/27, 9/7, 103/20, -13/5, 31/54, -19/2835], 
 'offsets': [-2, 1, 0, 2, 3, 4, 7], 
 'accuracy': 5}

Matrix Representation

For a given differential operator, you can get the matrix representation using the matrix(shape) method, e.g. for a small 1D grid of 10 points:

d2_dx2 = Diff(0, dx)**2
mat = d2_dx2.matrix((10,))  # this method returns a scipy sparse matrix
print(mat.toarray())

has the output

[[ 2. -5.  4. -1.  0.  0.  0.]
 [ 1. -2.  1.  0.  0.  0.  0.]
 [ 0.  1. -2.  1.  0.  0.  0.]
 [ 0.  0.  1. -2.  1.  0.  0.]
 [ 0.  0.  0.  1. -2.  1.  0.]
 [ 0.  0.  0.  0.  1. -2.  1.]
 [ 0.  0.  0. -1.  4. -5.  2.]]

If you have periodic boundary conditions, the matrix looks like that:

d2_dx2 = Diff(0, dx, periodic=True)**2
mat = d2_dx2.matrix((10,))  # this method returns a scipy sparse matrix
print(mat.toarray())

[[-2.  1.  0.  0.  0.  0.  1.]
 [ 1. -2.  1.  0.  0.  0.  0.]
 [ 0.  1. -2.  1.  0.  0.  0.]
 [ 0.  0.  1. -2.  1.  0.  0.]
 [ 0.  0.  0.  1. -2.  1.  0.]
 [ 0.  0.  0.  0.  1. -2.  1.]
 [ 1.  0.  0.  0.  0.  1. -2.]]

Stencils

findiff uses standard stencils (patterns of grid points) to evaluate the derivative. However, you can design your own stencil. A picture says more than a thousand words, so look at the following example for a standard second order accurate stencil for the 2D Laplacian $\displaystyle \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$:

This can be reproduced by findiff writing

offsets = [(0, 0), (1, 0), (-1, 0), (0, 1), (0, -1)]
stencil = Stencil(offsets, partials={(2, 0): 1, (0, 2): 1}, spacings=(1, 1))

The attribute stencil.values contains the coefficients

{(0, 0): -4.0, (1, 0): 1.0, (-1, 0): 1.0, (0, 1): 1.0, (0, -1): 1.0}

Now for a some more exotic stencil. Consider this one:

With findiff you can get it easily:

offsets = [(0, 0), (1, 1), (-1, -1), (1, -1), (-1, 1)]
stencil = Stencil(offsets, partials={(2, 0): 1, (0, 2): 1}, spacings=(1, 1))
stencil.values

which returns

{(0, 0): -2.0, (1, 1): 0.5, (-1, -1): 0.5, (1, -1): 0.5, (-1, 1): 0.5}

Symbolic Representations

As of version 0.10, findiff can also provide a symbolic representation of finite difference schemes suitable for using in conjunction with sympy. The main use case is to facilitate deriving your own iteration schemes.

from findiff import SymbolicMesh, SymbolicDiff

mesh = SymbolicMesh("x, y")
u = mesh.create_symbol("u")
d2_dx2, d2_dy2 = [SymbolicDiff(mesh, axis=k, degree=2) for k in range(2)]

(
    d2_dx2(u, at=(m, n), offsets=(-1, 0, 1)) + 
    d2_dy2(u, at=(m, n), offsets=(-1, 0, 1))
)

Outputs:

$$ \frac{u_{m,n + 1} + u_{m,n - 1} - 2 u_{m,n}}{\Delta y^2} + \frac{u_{m + 1,n} + u_{m - 1,n} - 2 u_{m,n}}{\Delta x^2} $$

Also see the example notebook.

Partial Differential Equations

findiff can be used to easily formulate and solve partial differential equation problems

$$ \mathcal{L}u(\vec{x}) = f(\vec{x}) $$

where $\mathcal{L}$ is a general linear differential operator.

In order to obtain a unique solution, Dirichlet, Neumann or more general boundary conditions can be applied.

Boundary Value Problems

Example 1: 1D forced harmonic oscillator with friction

Find the solution of

$$ \left( \frac{d^2}{dt^2} - \alpha \frac{d}{dt} + \omega^2 \right)u(t) = \sin{(2t)} $$

subject to the (Dirichlet) boundary conditions

$$ u(0) = 0, \hspace{1em} u(10) = 1 $$

from findiff import Diff, Id, PDE

shape = (300, )
t = numpy.linspace(0, 10, shape[0])
dt = t[1]-t[0]

L = Diff(0, dt)**2 - Diff(0, dt) + 5 * Id()
f = numpy.cos(2*t)

bc = BoundaryConditions(shape)
bc[0] = 0
bc[-1] = 1

pde = PDE(L, f, bc)
u = pde.solve()

Result:

Example 2: 2D heat conduction

A plate with temperature profile given on one edge and zero heat flux across the other edges, i.e.

$$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) u(x,y) = f(x,y) $$

with Dirichlet boundary condition

$$ \begin{align*} u(x,0) &= 300 \\ u(1,y) &= 300 - 200y \end{align*} $$

and Neumann boundary conditions

$$ \begin{align*} \frac{\partial u}{\partial x} &= 0, & \text{ for } x = 0 \\ \frac{\partial u}{\partial y} &= 0, & \text{ for } y = 0 \end{align*} $$

shape = (100, 100)
x, y = np.linspace(0, 1, shape[0]), np.linspace(0, 1, shape[1])
dx, dy = x[1]-x[0], y[1]-y[0]
X, Y = np.meshgrid(x, y, indexing='ij')

L = FinDiff(0, dx, 2) + FinDiff(1, dy, 2)
f = np.zeros(shape)

bc = BoundaryConditions(shape)
bc[1,:] = FinDiff(0, dx, 1), 0  # Neumann BC
bc[-1,:] = 300. - 200*Y   # Dirichlet BC
bc[:, 0] = 300.   # Dirichlet BC
bc[1:-1, -1] = FinDiff(1, dy, 1), 0  # Neumann BC

pde = PDE(L, f, bc)
u = pde.solve()

Result:

Citations

You have used findiff in a publication? Here is how you can cite it:

M. Baer. findiff software package. URL: https://github.com/maroba/findiff. 2018

BibTeX entry:

@misc{findiff,
  title = {{findiff} Software Package},
  author = {M. Baer},
  url = {https://github.com/maroba/findiff},
  key = {findiff},
  note = {\url{https://github.com/maroba/findiff}},
  year = {2018}
}

Development

Set up development environment

• Fork the repository
• Clone your fork to your machine
• Install in development mode:

pip install -e .

Running tests

From the console:

pip install pytest
pytest tests