Basic Derivatives

findiff works in any number of dimensions. Here we demonstrate the basics with 1D and 3D examples on uniform (equidistant) grids.

Setup

import numpy as np
from findiff import Diff, coefficients

Simple 1D Case

Suppose we want to compute the second derivative of \(f(x) = \sin(x)\) and \(g(x) = \cos(x)\):

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

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

Construct the differential operator \(\frac{d^2}{dx^2}\):

d2_dx2 = Diff(0, dx) ** 2

Apply it:

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

That’s it! The result arrays have the same shape as the input and contain the values of the second derivatives.

Finite Difference Coefficients

By default Diff uses second order accuracy. You can inspect the 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])}}

Higher accuracy orders are easy:

coefficients(deriv=2, acc=10)
d2_dx2 = Diff(0, dx, acc=10) ** 2

Simple 3D Case

Differentiate \(f(x, y, z) = \sin(x) \cos(y) \sin(z)\):

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)

Partial derivatives:

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

Mixed partial derivative \(\frac{\partial^2 f}{\partial x \partial y}\):

d2_dxdy = Diff(0, dx) * Diff(1, dy)
result = d2_dxdy(f)

General Linear Differential Operators

Diff objects can be added and multiplied by numbers or arrays:

# Constant coefficients
linear_op = Diff(0, dx)**2 + 2 * Diff(0, dx) * Diff(1, dy) + Diff(1, dy)**2

# Variable coefficients
linear_op = X * Diff(0, dx) + Y**2 * Diff(1, dy)

result = linear_op(f)