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Overview

The basis module provides spectral basis classes for discretizing differential equations. It includes polynomial bases (Jacobi, Chebyshev, Legendre), Fourier bases, and specialized multidimensional bases for disks, spheres, and shells.

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Interval Bases

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Chebyshev

ChebyshevT(coord, size, bounds, dealias=1)
Chebyshev polynomial basis of the first kind (T_n).

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Parameters

  • coord (Coordinate): Coordinate object
  • size (int): Number of modes
  • bounds (tuple): Domain bounds (lower, upper)
  • dealias (float or tuple, optional): Dealiasing scale factor (default: 1)

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Example

import dedalus.public as d3
import numpy as np

coord = d3.Coordinate('x')
basis = d3.ChebyshevT(coord, size=64, bounds=(0, 1))

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Legendre

Legendre(coord, size, bounds, dealias=1)
Legendre polynomial basis.

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Parameters

  • coord (Coordinate): Coordinate object
  • size (int): Number of modes
  • bounds (tuple): Domain bounds
  • dealias (float or tuple, optional): Dealiasing scale (default: 1)

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Example

import dedalus.public as d3

coord = d3.Coordinate('z')
basis = d3.Legendre(coord, size=128, bounds=(-1, 1))

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Jacobi

Jacobi(coord, size, bounds, a, b, a0=None, b0=None, dealias=1, library=None)
General Jacobi polynomial basis P_n^(a,b).

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Parameters

  • coord (Coordinate): Coordinate object
  • size (int): Number of modes
  • bounds (tuple): Domain bounds
  • a (float): Left endpoint parameter
  • b (float): Right endpoint parameter
  • a0 (float, optional): Grid left parameter (default: a)
  • b0 (float, optional): Grid right parameter (default: b)
  • dealias (float, optional): Dealiasing scale
  • library (str, optional): Transform library (‘matrix’ or ‘fftw_dct’)

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Example

import dedalus.public as d3

coord = d3.Coordinate('x')
# Jacobi(1/2, 1/2) = Ultraspherical(1) = Chebyshev(2nd kind)
basis = d3.Jacobi(coord, size=64, bounds=(0, 1), a=0.5, b=0.5)

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Fourier

ComplexFourier(coord, size, bounds, dealias=1)
RealFourier(coord, size, bounds, dealias=1)
Fourier basis for periodic domains.

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Parameters

  • coord (Coordinate): Coordinate object
  • size (int): Number of modes
  • bounds (tuple): Domain bounds
  • dealias (float, optional): Dealiasing scale

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Notes

  • ComplexFourier: Uses complex exponentials, supports complex fields
  • RealFourier: Uses sines/cosines, for real-valued fields only

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Example

import dedalus.public as d3
import numpy as np

coord = d3.Coordinate('x')  
# Complex Fourier for general use
basis = d3.ComplexFourier(coord, size=128, bounds=(0, 2*np.pi))

# Real Fourier for real-valued problems
basis = d3.RealFourier(coord, size=128, bounds=(0, 2*np.pi))

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Multidimensional Bases

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SphereBasis

SphereBasis(coordsys, shape, radius=1, dealias=(1,1), dtype=np.complex128)
Spherical harmonic basis on the sphere.

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Parameters

  • coordsys (S2Coordinates): S2 coordinate system
  • shape (tuple): Resolution (Nφ, Nθ)
  • radius (float, optional): Sphere radius (default: 1)
  • dealias (tuple, optional): Dealiasing scales
  • dtype: Data type

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Example

import dedalus.public as d3
import numpy as np

coords = d3.S2Coordinates('phi', 'theta')
basis = d3.SphereBasis(coords, (64, 32), radius=1, dtype=np.complex128)

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BallBasis

BallBasis(coordsys, shape, radius=1, dealias=(1,1,1), dtype=np.complex128)
Basis for the interior of a 3D ball.

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Parameters

  • coordsys (SphericalCoordinates): Spherical coordinate system
  • shape (tuple): Resolution (Nφ, Nθ, Nr)
  • radius (float, optional): Ball radius
  • dealias (tuple, optional): Dealiasing scales
  • dtype: Data type

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Example

import dedalus.public as d3
import numpy as np

coords = d3.SphericalCoordinates('phi', 'theta', 'r')
basis = d3.BallBasis(coords, (32, 16, 24), radius=1)

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ShellBasis

ShellBasis(coordsys, shape, radii=(0.5, 1), dealias=(1,1,1), dtype=np.complex128)
Basis for a spherical shell.

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Parameters

  • coordsys (SphericalCoordinates): Spherical coordinates
  • shape (tuple): Resolution (Nφ, Nθ, Nr)
  • radii (tuple, optional): Inner and outer radii
  • dealias (tuple, optional): Dealiasing scales
  • dtype: Data type

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Example

import dedalus.public as d3

coords = d3.SphericalCoordinates('phi', 'theta', 'r')
basis = d3.ShellBasis(coords, (64, 32, 48), radii=(0.5, 1.0))

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DiskBasis

DiskBasis(coordsys, shape, radius=1, dealias=(1,1), dtype=np.complex128)
Basis for a 2D disk.

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Parameters

  • coordsys (PolarCoordinates): Polar coordinate system
  • shape (tuple): Resolution (Nφ, Nr)
  • radius (float, optional): Disk radius
  • dealias (tuple, optional): Dealiasing scales
  • dtype: Data type

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Example

import dedalus.public as d3

coords = d3.PolarCoordinates('phi', 'r')
basis = d3.DiskBasis(coords, (64, 32), radius=2.0)

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AnnulusBasis

AnnulusBasis(coordsys, shape, radii=(0.5, 1), dealias=(1,1), dtype=np.complex128)
Basis for an annulus (2D shell).

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Parameters

  • coordsys (PolarCoordinates): Polar coordinates
  • shape (tuple): Resolution (Nφ, Nr)
  • radii (tuple, optional): Inner and outer radii
  • dealias (tuple, optional): Dealiasing scales
  • dtype: Data type

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Example

import dedalus.public as d3

coords = d3.PolarCoordinates('phi', 'r')  
basis = d3.AnnulusBasis(coords, (128, 64), radii=(1.0, 2.0))

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See Also

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