README.md

Python-Curl-Function

An extension to numpy using discrete fourier transforms to compute the curl of 2D and 3D functions. This produces results far more accurate than using 10th-order finite difference derivatives (which is as far as I tested).

Main Functions

curl2D computes the curl for a 2D input function $\boldsymbol{F} = (F_x, F_y)$.

curl3D computes the curl for a 3D input function $\boldsymbol{F} = (F_x, F_y, F_z)$.

rcurl2D computes the curl for a 2D purely real input function $\boldsymbol{F} = (F_x, F_y)$.

rcurl3D computes the curl for a 3D purely real input function $\boldsymbol{F} = (F_x, F_y, F_z)$.

Download

You can download it simply with the download button, or using

wget https://raw.githubusercontent.com/morgannamaths/Python-Curl-Function/main/python_curl_function.py

in the terminal.

Usage

Download the python_curl_function.py file and store it in the same file as your project (or wherever, really), and import it into your code using from python_curl_function import CurlFunction.

Explanation

Mathematically, the curl $\nabla \times \boldsymbol{F}$ can be written as

$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ F_x(x, y, z) & F_y(x, y, z) & F_z(x, y, z) \end{vmatrix}$

which can be further expanded to

$= \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\hat{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right)\hat{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\hat{k}$

Clearly, there are 6 derivatives (2 in the 2D case) required to compute. This code utilises the discrete fourier transform to compute these, which (based on testing) are more accurate than at least 10th-order finite difference derivatives, at the cost of being more computationally expensive.

$\hat{f}(\vec{k}) = \frac{1}{N} \Sigma_{m = 0}^{N-1} f(\vec{x})$ $e^{-i\vec{k} \bullet \vec{x}_m}$

$f(\vec{x}) = \Sigma_{k = -N/2 + 1}^{N/2} \hat{f}(\vec{k})$ $e^{i\vec{k} \bullet \vec{x}_m}$

where $\vec{x}_m$ = $\frac{2 \pi m}{N}$

Further Optimisation

For real functions, you can instead use rcurl2D and rcurl3D to perform a real discrete fourier transform. This is because the discrete fourier transform of a real function is Hermitian-symmetric and thus we can discard half of the transformed values to save on memory and computation.