`dedalus.core.basis`

Abstract and built-in classes for spectral bases.

Module Contents

logger

DEFAULT_LIBRARY

FFTW_RIGOR

DIRICHLET_PRECONDITIONING

class Basis

Base class for spectral bases.

These classes define methods for transforming, differentiating, and integrating corresponding series represented by their spectral coefficients.

Parameters

base_grid_size (int) – Number of grid points
interval (tuple of floats) – Spatial domain of basis
dealias (float, optional) – Fraction of modes to keep after dealiasing (default: 1.)

Variables

grid_dtype (dtype) – Grid data type
coeff_size (int) – Number of spectral coefficients
coeff_embed (int) – Padded number of spectral coefficients for transform
coeff_dtype (dtype) – Coefficient data type

property library

abstract set_dtype(grid_dtype): Set transforms based on grid data type.

abstract forward(gdata, cdata, axis, meta, scale): Grid-to-coefficient transform.

abstract backward(cdata, gdata, axis, meta, scale): Coefficient-to-grid transform.

abstract differentiate(cdata, cderiv, axis): Differentiate using coefficients.

abstract integrate(cdata, cint, axis): Integrate over interval using coefficients.

abstract interpolate(cdata, cint, position, axis): Interpolate in interval using coefficients.

grid_size(scale): Compute scaled grid size.

grid_spacing(scale=1.0): Compute grid spacings.

grid_array_object(domain, axis): Grid array object.

grid_spacing_object(domain, axis)

check_arrays(cdata, gdata, axis, scale): Verify provided arrays sizes and dtypes are correct. Build compliant arrays if not provided.

class TransverseBasis: Base class for bases supporting transverse differentiation.

class ImplicitBasis

Base class for bases supporting implicit methods.

These bases define the following matrices encoding the respective linear functions acting on a series represented by its spectral coefficients:

Linear operators (square matrices):: Pre : preconditioning (default: identity) Diff : differentiation Mult(p) : multiplication by p-th basis element
Linear functionals (vectors):: left_vector : left-endpoint evaluation right_vector : right-endpoint evaluation integ_vector : integration over interval interp_vector : interpolation in interval

abstract Multiply(p, ncc_basis_meta, arg_basis_meta): p-element multiplication matrix.

abstract Precondition(): Preconditioning matrix.

DropTau(n_tau): Matrix dropping tau+match rows.

DropNonfirst(): Matrix dropping non-first rows.

DropNonconstant(): Matrix dropping non-constant rows.

DropMatch(): Matrix dropping match rows.

PreconditionDropTau(n_tau): Preconditioning with tau+match filtering.

PreconditionDropMatch(): Preconditioning with match filtering.

NCC(ncc_basis_meta, arg_basis_meta, coeffs, cutoff, max_terms): Build NCC multiplication matrix.

class Chebyshev(name, base_grid_size, interval=(- 1, 1), dealias=1, tau_after_pre=True)

Chebyshev polynomial basis on the roots grid.

element_label = 'T'

separable = False

coupled = True

default_meta()

grid(scale=1.0): Build Chebyshev roots grid.

grid_spacing(scale=1.0): Build Chebyshev roots grid spacing.

set_dtype(grid_dtype): Determine coefficient properties from grid dtype.

Integrate(): Build integration class.

Interpolate(): Buld interpolation class.

Differentiate(): Build differentiation class.

Precondition()

Preconditioning matrix.

T_n = (U_n - U_(n-2)) / 2 U_(-n) = -U_(n-2)

Dirichlet()

Dirichlet recombination matrix.

D[0] = T[0] D[1] = T[1] D[n] = T[n] - T[n-2]

<T[i]|D[j]> = <T[i]|T[j]> - <T[i]|T[j-2]>: = δ(i,j) - δ(i,j-2)

Multiply(p, ncc_basis_meta, arg_basis_meta): p-element multiplication matrix.

class Legendre(name, base_grid_size, interval=(- 1, 1), dealias=1, tau_after_pre=True)

Legendre polynomial basis on the roots grid.

element_label = 'P'

separable = False

coupled = True

default_meta()

grid(scale=1.0): Build Legendre polynomial roots grid.

set_dtype(grid_dtype): Determine coefficient properties from grid dtype.

Integrate(): Build integration class.

Interpolate(): Buld interpolation class.

Differentiate(): Build differentiation class.

Precondition()

Preconditioning matrix.: 2 * (2*n + 1) * P[n] = (n + 2) * J11[n] - n * J11[n-2]

Dirichlet()

Dirichlet recombination matrix.

D[0] = P[0] D[1] = P[1] D[n] = P[n] - P[n-2]

<P[i]|D[j]> = <P[i]|T[j]> - <P[i]|T[j-2]>: = δ(i,j) - δ(i,j-2)

Multiply(p, ncc_basis_meta, arg_basis_meta): p-element multiplication matrix.

class Hermite(name, base_grid_size, center=0.0, stretch=1.0, dealias=1, tau_after_pre=False)

Hermite function/polynomial basis.

Interval:: The functions live on (-inf, inf) but are centered and scaled by the affine transformation from (-1, 1) to the specified interval.
Hermite functions (with envelope, default):: hf[n](xp) = exp(-xn^2/2) H[n](xn) / N[n] N[n]**2 = π^(1/2) 2^n n! integ hf[n] hf[m] dxn = δ[n,m]
Hermite polynomials (without envelope):: hp[n](xp) = H[n](xn) integ hp[n] hp[m] exp(-xn**2) dx = δ[n,m] N[n]**2

element_label = 'h'

separable = False

coupled = True

default_meta()

grid(scale=1.0): Build Hermite polynomial roots grid.

grid_array_object(domain, axis): Grid array object.

set_dtype(grid_dtype): Determine coefficient properties from grid dtype.

Integrate(): Build integration class.

Interpolate(): Buld interpolation class.

Differentiate(): Build differentiation class.

Precondition(): Preconditioning matrix.

Dirichlet(): Dirichlet recombination matrix.

Multiply(p, ncc_basis_meta, arg_basis_meta): Hermite multiplication matrix.

class Laguerre(name, base_grid_size, edge=0.0, stretch=1.0, dealias=1, tau_after_pre=False)

Laguerre function/polynomial basis.

Interval:: The functions live on (0, inf) but are centered and scaled by the affine transformation from (0, 1) to the specified interval.
Laguerre functions (with envelope, default):: gn[n](xp) = exp(-x/2) L[n](xn) integ gn[n] gn[m] dxn = δ[n,m]
Laguerre polynomials (without envelope):: gp[n](xp) = L[n](xn) integ gp[n] gp[m] exp(-xn) dx = δ[n,m]

element_label = 'g'

separable = False

coupled = True

default_meta()

grid(scale=1.0): Build Laguerre polynomial roots grid.

grid_array_object(domain, axis): Grid array object.

set_dtype(grid_dtype): Determine coefficient properties from grid dtype.

Integrate(): Build integration class.

Interpolate(): Buld interpolation class.

Differentiate(): Build differentiation class.

Precondition()

Preconditioning matrix.: L[n;1] = sum_{i=0}^{n} L[n;0] L[n;0] = L[n;1] - L[n-1;1]

Dirichlet()

Dirichlet recombination matrix.

G[0] = g[0] G[n] = g[n] - g[n-1]

<g[i]|G[j]> = <g[i]|g[j]> - <g[i]|g[j-1]>: = δ(i,j) - δ(i,j-1)

Multiply(p, ncc_basis_meta, arg_basis_meta): Laguerre multiplication matrix.

class Fourier(name, base_grid_size, interval=(0, 2 * pi), dealias=1)

Fourier complex exponential basis.

separable = True

coupled = False

element_label = 'k'

default_meta()

grid(scale=1.0): Build evenly spaced Fourier grid.

grid_spacing(scale=1.0): Build Fourier grid spacing.

set_dtype(grid_dtype): Determine coefficient properties from grid dtype.

Integrate(): Build integration class.

Interpolate(): Build interpolation class.

Differentiate(): Build differentiation class.

HilbertTransform(): Build Hilbert transform class.

class SinCos(name, base_grid_size, interval=(0, pi), dealias=1)

Sin/Cos series basis.

element_label = 'k'

separable = True

coupled = False

default_meta()

grid(scale=1.0): Evenly spaced interior grid: cos(Nx) = 0

grid_spacing(scale=1.0): Build cosine grid spacing.

grid_array_object(domain, axis): Grid array object.

set_dtype(grid_dtype): Determine coefficient properties from grid dtype.

Integrate(): Build integration class.

Interpolate(): Build interpolation class.

Differentiate(): Build differentiation class.

HilbertTransform(): Build Hilbert transform class.

class Compound(name, subbases, dealias=1)

Compound basis joining adjascent subbases.

property library

separable = False

coupled = True

default_meta()

grid(scale=1.0): Build compound grid.

grid_spacing(scale=1.0): Build compound grid spacing.

grid_array_object(domain, axis): Grid array object.

set_dtype(grid_dtype): Determine coefficient properties from grid dtype.

coeff_start(index)

grid_start(index, scale)

sub_gdata(gdata, index, axis): Retreive gdata corresponding to one subbasis.

sub_cdata(cdata, index, axis): Retrieve cdata corresponding to one subbasis.

forward(gdata, cdata, axis, meta, scale): Forward transforms.

backward(cdata, gdata, axis, meta, scale): Backward transforms.

Integrate(): Build integration class.

Interpolate(): Buld interpolation class.

Differentiate(): Build differentiation class.

Precondition(): Preconditioning matrix.

Dirichlet()

Multiply(subindex, p, ncc_basis_meta, arg_basis_meta): p-element multiplication matrix.

NCC(ncc_basis_meta, arg_basis_meta, coeffs, cutoff, max_terms): Build NCC multiplication matrix.

DropTau(n_tau): Matrix dropping tau+match rows.

DropNonfirst(): Matrix dropping non-first rows.

DropNonconstant(): Matrix dropping non-constant rows.

DropMatch(): Matrix dropping last row from each subbasis except the last.

PreconditionDropTau(n_tau): Preconditioning and tau+match filtering.

PreconditionDropMatch(): Preconditioning and match filtering.

MatchRows(): Matrix matching subbases.

dedalus.core.basis

Module Contents

`dedalus.core.basis`