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PyMPDATA

PyMPDATA is a high-performance Numba-accelerated Pythonic implementation of the MPDATA algorithm of Smolarkiewicz et al. for numerically solving generalised transport equations - partial differential equations used to model conservation/balance laws, scalar-transport problems, convection-diffusion phenomena (in geophysical fluid dynamics and beyond). As of the current version, PyMPDATA supports homogeneous transport in 1D, 2D and 3D using structured meshes, optionally generalised by employment of a Jacobian of coordinate transformation. PyMPDATA includes implementation of a set of MPDATA variants including flux-corrected transport (FCT), infinite-gauge, divergent-flow and third-order-terms options. It also features support for integration of Fickian-terms in advection-diffusion problems using the pseudo-transport velocity approach. In 2D and 3D simulations, domain-decomposition is used for multi-threaded parallelism.

PyMPDATA is engineered purely in Python targeting both performance and usability, the latter encompassing research users', developers' and maintainers' perspectives. From researcher's perspective, PyMPDATA offers hassle-free installation on multitude of platforms including Linux, OSX and Windows, and eliminates compilation stage from the perspective of the user. From developers' and maintainers' perspective, PyMPDATA offers wide unit-test coverage, multi-platform continuous integration setup, seamless integration with Python debugging and profiling tools, and robust susceptibility to static code analysis within integrated development environments.

PyMPDATA design features a custom-built multi-dimensional Arakawa-C grid layer allowing to concisely represent multi-dimensional stencil operations. The grid layer is built on top of NumPy's ndarrays (using "C" ordering) using Numba's @njit functionality for high-performance array traversals. It enables one to code once for multiple dimensions, and automatically handles (and hides from the user) any halo-filling logic related with boundary conditions.

PyMPDATA ships with a set of examples/demos offered as github-hosted Jupyer notebooks offering single-click deployment in the cloud using mybinder.org or using colab.research.google.com. The examples/demos reproduce results from several published works on MPDATA and its applications, and provide a validation of the implementation and its performance.

Dependencies and installation

PyMPDATA has Python-only dependencies, all available through PyPi and listed in the project's requirements.txt file.

To install PyMPDATA, one may use:

pip3 install --pre git+https://github.com/atmos-cloud-sim-uj/PyMPDATA.git

Examples/Demos:

PyMPDATA ships with several demos that reproduce results from the literature, including:

  • Smolarkiewicz 2006 Figs 3,4,10,11 & 12 Binder Open In Colab
    (1D homogeneous cases depicting infinite-gauge and flux-corrected transport cases)
  • Arabas & Farhat 2020 Figs 1-3 & Tab. 1 Binder Open In Colab
    (1D advection-diffusion example based on Black-Scholes equation)
  • Olesik et al. 2020 (in preparation) Binder Open In Colab
    (1D particle population condensational growth problem with coordinate transformations)
  • Molenkamp test (as in Jaruga et al. 2015, Fig. 12) Binder
    (2D solid-body rotation test)
  • 1D advection-diffusion example with animation Binder Open In Colab

Package structure and API:

PyMPDATA is designed as a simple library in the spirit of "Do One Thing And Do It Well", namely to numerically solve the following equation:

\partial_t (G \psi) + \nabla \cdot (Gu \psi) = 0

where scalar field \psi is referred to as the advectee, vector field u is referred to as advector, and the G factor corresponds to optional coordinate transformation.

The key classes constituting the PyMPDATA interface are summarised below:

Options class

The Options class groups both algorithm variant options as well as some implementation-related flags that need to be set at the first place. All are set at the time of instantiation using the following keyword arguments of the constructor (all having default values indicated below):

  • n_iters:int = 2: number of iterations (2 means upwind + one corrective iteration)
  • infinite_gauge: bool = False: flag enabling the infinite-gauge option (does not maintain sign of the advected field, thus in practice implies switching flux corrected transport on)
  • divergent_flow: bool = False: flag enabling divergent-flow terms when calculating antidiffusive velocity
  • flux_corrected_transport: bool = False: flag enabling flux-corrected transport (FCT) logic (a.k.a. non-oscillatory or monotone variant)
  • third_order_terms: bool = False: flag enabling third-order terms
  • epsilon: float = 1e-15: value added to potentially zero-valued denominators
  • non_zero_mu_coeff: bool = False: flag indicating if code for handling the Fickian term is to be optimised out

For a discussion of the above options, see e.g., Smolarkiewicz & Margolin 1998.

In most use cases of PyMPDATA, the first thing to do is to instantiate the Options class with arguments suiting the problem at hand, e.g.:

from PyMPDATA import Options
options = Options(n_iters=3, infinite_gauge=True, flux_corrected_transport=True)

Arakawa-C grid layer and boundary conditions

The arakawa_c subpackage contains modules implementing the Arakawa-C staggered grid in which:

  • scalar fields are discretised onto cell-center points,
  • vector fields are discretised onto cell-boundary points.

In PyMPDATA, the solution domain is assumed to extend from the first cell's boundary to the last cell's boundary (thus first scalar field value is at [\Delta x/2, \Delta y/2]).

From the user perspective, the two key classes with their init methods are:

The data parameters are expected to be Numpy arrays or tuples of Numpy arrays, respectively. The halo parameter is the extent of ghost-cell region that will surround the data and will be used to implement boundary conditions. Its value (in practice 1 or 2) is dependent on maximal stencil extent for the MPDATA variant used and can be easily obtained using the Options.n_halo property.

As an example, the code below shows how to instantiate a scalar and a vector field given a 2D constant-velocity problem, using a grid of 100x100 points and cyclic boundary conditions (with all values set to zero):

from PyMPDATA import ScalarField
from PyMPDATA import VectorField
from PyMPDATA import PeriodicBoundaryCondition
import numpy as np

nx, ny = 100, 100
halo = options.n_halo
advectee = ScalarField(
    data=np.zeros((nx, ny)), 
    halo=halo, 
    boundary_conditions=(PeriodicBoundaryCondition(), PeriodicBoundaryCondition())
)
advector = VectorField(
    data=(np.zeros((nx+1, ny)), np.zeros((nx, ny+1))),
    halo=halo,
    boundary_conditions=(PeriodicBoundaryCondition(), PeriodicBoundaryCondition())    
)

Note that the shapes of arrays representing components of the velocity field are different than the shape of the scalar field array due to employment of the staggered grid.

Besides the exemplified Cyclic class representing periodic boundary conditions, PyMPDATA supports Extrapolated and Constant boundary conditions.

Stepper

The logic of the MPDATA iterative solver is represented in PyMPDATA by the Stepper class.

When instantiating the Stepper, the user has a choice of either supplying just the number of dimensions or specialising the stepper for a given grid:

from PyMPDATA import Stepper

stepper = Stepper(options=options, n_dims=2)

or

stepper = Stepper(options=options, grid=(nx, ny))

In the latter case, noticeably faster execution can be expected, however the resultant stepper is less versatile as bound to the given grid size. If number of dimensions is supplied only, the integration will take longer, yet same instance of the stepper can be used for different grids.

Since creating an instance of the Stepper class involves time consuming compilation of the algorithm code, the class is equipped with a cache logic - subsequent calls with same arguments return references to previously instantiated objects. Instances of Stepper contain no mutable data and are (thread-)safe to be reused.

The init method of Stepper has an optional non_unit_g_factor argument which is a Boolean flag enabling handling of the G factor term which can be used to represent coordinate transformations and/or variable fluid density.

Optionally, the number of threads to use for domain decomposition in first (non-contiguous) dimension during 2D and 3D calculations may be specified using the optional n_threads argument with a default value of numba.get_num_threads(). The multi-threaded logic of PyMPDATA depends thus on settings of numba, namely on the selected threading layer (either via NUMBA_THREADING_LAYER env var or via numba.config.THREADING_LAYER) and the selected size of the thread pool (NUMBA_NUM_THREADS env var or numba.config.NUMBA_NUM_THREADS).

Solver

Instances of the Solver class are used to control the integration and access solution data. During instantiation, additional memory required by the solver is allocated according to the options provided.

The only method of the Solver class besides the init is advance(self, nt: int, mu_coeff: float = 0) which advances the solution by nt timesteps, optionally taking into account a given value of diffusion coefficient.

Solution state is accessible through the Solver.advectee property.

Continuing with the above code snippets, instantiating a solver and making one integration step looks as follows:

from PyMPDATA import Solver
solver = Solver(stepper=stepper, advectee=advectee, advector=advector)
solver.advance(nt=1)
state = solver.advectee.get()

Debugging

PyMPDATA relies heavily on Numba to provide high-performance number crunching operations. Arguably, one of the key advantage of embracing Numba is that it can be easily switched off. This brings multiple-order-of-magnitude drop in performance, yet it also make the entire code of the library susceptible to interactive debugging, one way of enabling it is by setting the following environment variable before importing PyMPDATA:

import os
os.environ["NUMBA_DISABLE_JIT"] = "1"

Credits:

Development of PyMPDATA is supported by the EU through a grant of the Foundation for Polish Science (POIR.04.04.00-00-5E1C/18).

copyright: Jagiellonian University
licence: GPL v3

Other open-source MPDATA implementations: