numgrids

Working with numerical grids made easy.

Main Features

• Quickly define numerical grids for any rectangular or curvilinear coordinate system
• Differentiation and integration
• Interpolation
• Easy manipulation of meshed functions
• Using high precision spectral methods (FFT + Chebyshev) wherever possible
• Includes multigrid functionality
• Fully compatible with numpy

Installation

pip install --upgrade numgrids

Quick Start

As a quick example, here is how you define a grid on the unit disk using polar coordinates. Along the azimuthal (angular) direction, choose an equidistant spacing with periodic boundary conditions:

from numgrids import *
from numpy import pi

axis_phi = Axis(AxisType.EQUIDISTANT, 50, 0, 2*pi, periodic=True)

Along the radial axis, let's choose a non-equidistant spacing:

axis_radial = Axis(AxisType.CHEBYSHEV, 20, 0, 1)

Now combine the axes to a grid:

grid = Grid(axis_radial, axis_phi)

Sample a meshed function on this grid:

from numpy import exp, sin

R, Phi = grid.meshed_coords
f = R**2 * sin(Phi)**2

Define partial derivatives $\partial/\partial r$ and $\partial/\partial \varphi$ and apply them:

# second argument means derivative order, third argument means axis index:
d_dr = Diff(grid, 1, 0) 
d_dphi = Diff(grid, 1, 1)

df_dr = d_dr(f)
df_dphi = d_dphi(f)

Obtain the matrix representation of the differential operators:

d_dr.as_matrix()

Out: <1000x1000 sparse matrix of type '<class 'numpy.float64'>'
	with 20000 stored elements in COOrdinate format>

Define integration operator

$$ \int \dots dr d\varphi $$

I = Integral(grid)

Calculate the area integral

$$ \int f(r, \varphi) r dr d\varphi $$

(taking into account the appropriate integration measure $r$ for polar coordinates):

I(f * R)

Setting boundary values to zero

f[grid.boundary] = 0  # grid.boundary is boolean mask selecting boundary grid points

or to something more complicated:

f[grid.boundary] = exp(-R[grid.boundary])

Create an interpolation function

inter = Interpolator(grid, f)

Interpolate for a single point

point = (0.1, 0.5)
inter(point)

or for many points at once, like for a parametrized curve:

t = np.linspace(0, 1, 100)
points = zip(2*t, t**2)
inter(points)

Usage / Example Notebooks

To get an idea how numgrids can be used, have a look at the following example notebooks:

• How to define grids
• Partial derivatives in any dimension
• Polar coordinates on unit disk
• Spherical Grid and the Spherical Laplacian
• Solving the Schrödinger equation for the quantum harmonic oscillator

Development

Setting up the project

Clone the repository

git clone https://github.com/maroba/numgrids.git

In the project root directory, submit

python setup.py develop

to install the package in development mode.

Run the tests:

python -m unittest discover tests

Contributing

1. Fork the repository
2. Develop
3. Write tests!
4. Create an issue
5. Create a pull request, when done