Basic Examples of findiff¶

findiff works in any dimension. But for the sake of demonstration, let’s concentrate on the cases 1D and 3D. We are using uniform, i.e. equidistant, grids here. The non-uniform case will be shown in another notebook.

Preliminaries¶

Our imports:

import numpy as np
from findiff import FinDiff, coefficients, Coefficient

Simple 1D Cases¶

Suppose we want to differentiate two 1D-arrays f and g, which are filled with values from a function

$f(x) = \sin(x) \quad \mbox{and}\quad g(x) = \cos(x)$

and we want to take the 2nd derivative. This is easy done analytically:

$\frac{d^2f}{dx^2} = -\sin(x) \quad \mbox{and}\quad \frac{d^2g}{dx^2} = -\cos(x)$

Let’s do this numerically with findiff. First we set up the grid and the arrays:

x = np.linspace(0, 10, 100)
dx = x[1] - x[0]
f = np.sin(x)
g = np.cos(x)

Then we construct the derivative object, which represents the differential operator $\frac{d^2}{dx^2}$ :

d2_dx2 = FinDiff(0, dx, 2)

The first parameter is the axis along which to take the derivative. Since we want to apply it to the one and only axis of the 1D array, this is a 0. The second parameter is the grid spacing, the third parameter the derivative order you want, in our case 2. If you want a first derivative, you can skip the third argument as it defaults to 1.

Then we apply the operator to f and g, respectively:

result_f = d2_dx2(f)
result_g = d2_dx2(g)

That’s it! The arrays result_fand result_g have the same shape as the arrays f and g and contain the values of the second derivatives.

Finite Difference Coefficients¶

By default the FinDiff class uses second order accuracy. For the second derivative, it uses the following finite difference coefficients:

coefficients(deriv=2, acc=2)

{'backward': {'coefficients': array([-1.,  4., -5.,  2.]),
  'offsets': array([-3, -2, -1,  0])},
 'center': {'coefficients': array([ 1., -2.,  1.]),
  'offsets': array([-1,  0,  1])},
 'forward': {'coefficients': array([ 2., -5.,  4., -1.]),
  'offsets': array([0, 1, 2, 3])}}

But FinDiff can handle any accuracy order. For instance, have you ever wondered, what the 10th order accurate coefficients look like? Here they are:

coefficients(deriv=2, acc=10)

{'backward': {'coefficients': array([  -0.53253968,    6.42373016,  -35.55158728,  119.41369042,
         -271.26190464,  439.39444427, -521.11333314,  457.02976176,
         -295.51984119,  138.59325394,  -44.43730158,    7.56162698]),
  'offsets': array([-11, -10,  -9,  -8,  -7,  -6,  -5,  -4,  -3,  -2,  -1,   0])},
 'center': {'coefficients': array([ 3.17460317e-04, -4.96031746e-03,  3.96825397e-02, -2.38095238e-01,
          1.66666667e+00, -2.92722222e+00,  1.66666667e+00, -2.38095238e-01,
          3.96825397e-02, -4.96031746e-03,  3.17460317e-04]),
  'offsets': array([-5, -4, -3, -2, -1,  0,  1,  2,  3,  4,  5])},
 'forward': {'coefficients': array([   7.56162876,  -44.43731776,  138.59331976, -295.52000468,
          457.03003946, -521.1136706 ,  439.39474213, -271.26209495,
          119.41377646,  -35.55161345,    6.42373497,   -0.53254009]),
  'offsets': array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11])}}

Accuracy order¶

If you want to use for example 10th order accuracy, just tell the FinDiff constructor to use it:

d2_dx2 = FinDiff(0, dx, 2, acc=10)
result = d2_dx2(f)

Simple 3D Cases¶

Now let’s differentiate a 3D-array f representing the function

$f(x, y, z) = \sin(x) \cos(y) \sin(z)$

x, y, z = [np.linspace(0, 10, 100)]*3
dx, dy, dz = x[1] - x[0], y[1] - y[0], z[1] - z[0]
X, Y, Z = np.meshgrid(x, y, z, indexing='ij')
f = np.sin(X) * np.cos(Y) * np.sin(Z)

The partial derivatives $\frac{\partial f}{\partial x}$ or $\frac{\partial f}{\partial z}$ are given by

d_dx = FinDiff(0, dx)
d_dz = FinDiff(2, dz)

The x-axis is the 0th axis, y, the first, z the 2nd, etc. The third mixed partial derivative $\frac{\partial^3 f}{\partial x^2 \partial y}$ is specified by two tuples as arguments, one for each partial derivative:

d3_dx2dy = FinDiff((0, dx, 2), (1, dy))
result = d3_dx2dy(f)

Of course, the accuracy order can be specified the same way as for 1D.

General Linear Differential Operators¶

FinDiff objects can bei added and easily multiplied by numbers. For example, to express

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

we can say

linear_op = FinDiff(0, dx, 2) + 2 * FinDiff((0, dx), (1, dy)) + FinDiff(1, dy, 2)

Variable Coefficients¶

If you want to multiply by variables instead of plain numbers, you have to encapsulate the variable in a Coefficient object. For example,

$x \frac{\partial}{\partial x} + y^2 \frac{\partial}{\partial y}$

is

linear_op = Coefficient(X) * FinDiff(0, dx) + Coefficient(Y**2) * FinDiff(1, dy)

Applying those general operators works the same way as for the simple derivatives:

result = linear_op(f)