PyMPDATA usage examples reproducing results from literature and depicting how to use PyMPDATA in Python from Jupyter notebooks
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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 $$\begin{eqnarray}\partial_t \vec{v}+\nabla\cdot\left(\vec{v}\otimes\vec{v}\right)=-\nabla\pi-\vec{g}\frac{\theta^\prime}{\theta_0} |
Jaruga_et_al_2015 |
| advection-diffusion equation $$ \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.
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