pympdata-examples 1.7.0

PyMPDATA usage examples reproducing results from literature and depicting how to use PyMPDATA in Python from Jupyter notebooks

Introduction

PyMPDATA examples are bundled with PyMPDATA and located in the examples subfolder. They constitute a separate PyMPDATA_examples Python package which is also available at PyPI. The examples have additional dependencies listed in PyMPDATA_examples package setup.py file. Running the examples requires the PyMPDATA_examples package to be installed.

We recommend you look through the example gallery below to see the examples in action.

Example gallery

Unless stated otherwise the following examples solve the basic advection equation: $$ \partial_t (\psi) + \nabla \cdot (u \psi) = 0 $$

The examples are grouped by the dimensionality of the computational grid.

in 1D

tags	link
advection-diffusion equation $$ \partial_t (\psi) + \nabla \cdot (u \psi) + \mu \Delta (\psi) = 0 $$	advection_diffusion_1d*
Black-Scholes equation, option pricing $$ \frac{\partial f}{\partial t} + rS \frac{\partial f}{\partial S} + \frac{\sigma^2}{2} S^2 \frac{\partial^2 f}{\partial S^2} - rf = 0$$	Arabas_and_Farhat_2020*
advection equation, homogeneous, several algorithm variants comparison: infinite-gauge, flux-corrected,..	Smolarkiewicz_2006_Figs_3_4_10_11_12
Size-spectral advection, particle population condensational growth, coordinate transformation $$ \partial_t (G \psi) + \nabla \cdot (Gu \psi) = 0 $$	Olesik_et_al_2022*
advection equation, double-pass donor-cell option	DPDC
Burgers' equation $$ \frac{\partial u}{\partial t} = -\frac{1}{2} \frac{\partial u^2}{\partial x} $$	burgers_equation*

in 2D

tags	link
Boussinesq system for buoyancy-driven flow $$\begin{eqnarray}\partial_t \vec{v}+\nabla\cdot\left(\vec{v}\otimes\vec{v}\right)=-\nabla\pi-\vec{g}\frac{\theta^\prime}{\theta_0}\\ \partial_t\theta+\nabla\cdot\left(\vec{v}\theta\right)=0\\ \nabla\cdot\vec{v}=0\end{eqnarray}$$	Jaruga_et_al_2015
advection-diffusion equation $$ \partial_t (\psi) + \nabla \cdot (u \psi) + \mu \Delta (\psi) = 0 $$	advection_diffusion_2d*
Spectral-spatial advection, particle population condensational growth in a vertical column of air, time dependent flow	Shipway_and_Hill_2012
shallow-water equations $$\begin{eqnarray} \partial_t h + \partial_x (uh) + \partial_y (vh) &=& 0~ \\ \partial_t (uh) + \partial_x ( uuh) + \partial_y (vuh) &=& - h \partial_x h~ \\ \partial_t (vh) + \partial_x ( uvh) + \partial_y (vvh) &=& - h \partial_y h~ \end{eqnarray}$$	Jarecka_et_al_2015*
advection equation, solid body rotation	Molenkamp_test_as_in_Jaruga_et_al_2015_Fig_12*
advection equation, solid body rotation	wikipedia_example*
advection equation, coordinate transformation, spherical coordinates, see also examples in PyMPDATA-MPI $$ \partial_t (G \psi) + \nabla \cdot (Gu \psi) = 0 $$	Williamson_and_Rasch_1989_as_in_Jaruga_et_al_2015_Fig_14
advection equation, comparison against DG solution using Trixi.jl (Ranocha et al. 2022)	trixi_comparison
Black-Scholes equation, option pricing $$ \frac{\partial f}{\partial t} + rS \frac{\partial f}{\partial S} + \frac{\sigma^2}{2} S^2 \frac{\partial^2 f}{\partial S^2} + \frac{dA}{dt}\frac{\partial f}{\partial A} - rf = 0$$	Magnuszewski_et_al_2025

in 3D

tags	link
homogeneous advection equation	Smolarkiewicz_1984
homogeneous advection equation, performance comparison against libmpdata++, scalability analysis in respect to threads	Bartman_et_al_2022

* - with comparison against analytic solution

Installation

Since the examples package includes Jupyter notebooks (and their execution requires write access), the suggested install and launch steps are:

git clone https://github.com/open-atmos/PyMPDATA-examples.git cd PyMPDATA-examples pip install -e . jupyter-notebook

Alternatively, one can also install the examples package from pypi.org by using pip install PyMPDATA-examples.