Tutorials & Examples

The following notebooks provide an overview of the functionality provided by NumDF including how to compute the CDF, PDF and QDF, how to compute compositions of these objects and how work with external data from numerical simulations. To run the notebooks, you will need to install jupyter inside your activated Firedrake virtualenv.

These notebooks are maintained in the NumDF repository, so all the material is available in your NumDF installation source directory while the notebooks are in the directory NumDF/docs/notebooks.

Thanks to the excellent FEM on Colab by Francesco Ballarin, you can run the notebooks on Google Colab through your web browser, without installing Firedrake or NumDF.

We also provide links to non-interactive renderings of the notebooks using Jupyter nbviewer.

Example 1 - Analytic functions

In this notebook, we demonstrate the basic functionality of NumDF including how to compute the CDF, PDF and QDF of a one and two dimensional analytical function. A rendered version of this notebook is available here and there is a version on Colab

Example 2 - Numerical functions

We then consider a more practically relevant example where the function specified is obtained as the output of a direct numerical simulation. Considering a two dimensional Kelvin-Helmholtz instability we present the time evolution of the CDF for which a rendered version of this notebook is available here and there is a version on Colab

Example 3 - Available potential energy

Next, we discuss how to compute the available potential energy for a simple two dimensional field and compare our methods estimate with an analytical calculation. This example builds on the previous example by requiring the integral of the composition of two CDFs to be evaluated. A rendered version of this notebook is available here and there is a version on Colab

Example 4 - Numerical convergence

Finally we show that the numerical implementation of our numerical method is consistent and discuss the challenges that arise when computing the density of common functions whose PDFs typically contain singularities. A rendered version of this notebook is available here and there is a version on Colab