Utility Functions

Spectral Differentiation

differentiate(f, domain, coord; order=1, filter=:none)

Spectrally differentiate a function given as physical-space values on the grid. This is the primary tool for computing derivatives of background profiles.

Chebyshev directions — converts to Chebyshev T coefficients, applies the ultraspherical differentiation chain, and converts back to physical space.

Fourier directions — applies (ik)^p in Fourier space with optional spectral filtering.

domain = Domain(z = Chebyshev(64, [0, 1]))
z = gridpoints(domain, :z)

U = @. z^2
dUdz   = differentiate(U, domain, :z)            # 2z
d2Udz2 = differentiate(U, domain, :z; order=2)   # 2

Fourier with filtering:

domain = Domain(y = Fourier(64, [0, 2*pi]))
y = gridpoints(domain, :y)
f = sin.(y)
dfdy = differentiate(f, domain, :y; filter=:exp)   # exponential filter

Eigenvalue Analysis

print_evals(lambdas)

Pretty-print eigenvalues in a formatted table.

sort_evals(lambdas, chi, by; rev=true)

Sort eigenvalues and eigenvectors. by can be a Symbol (:R, :I, :M) or String ("R", "I", "M").

lambdas_sorted, chi_sorted = sort_evals(lambdas, chi, :R; rev=true)

remove_evals(lambdas, chi, lower, higher, which)

Filter eigenvalues within bounds. which can be Symbol or String.

lambdas_filtered, chi_filtered = remove_evals(lambdas, chi, 0.1, 1e5, :R)

Grid and Transform Utilities

gridpoints(domain, dims...)

Return grid points for resolved dimensions.

z = gridpoints(domain, :z)              # 1D
y, z = gridpoints(domain, :y, :z)       # separate vectors

meshgrid(domain, dim1, dim2)

Return 2D meshgrid arrays (Ndim2 x Ndim1, z-fastest ordering).

Y, Z = meshgrid(domain, :y, :z)
U = @. Z - 0.5 + 0.1 * sin(2*pi * Y)  # 2D field
prob[:U] = vec(U)

to_coefficients(f, coord_type)

Transform physical-space values to spectral coefficients.

c = to_coefficients(f_values, :chebyshev)
c = to_coefficients(f_values, :fourier)

to_physical(c, coord_type; x=nothing)

Transform spectral coefficients back to physical space.

chebyshev_points(N, a, b)

N Chebyshev-Gauss-Lobatto points on [a, b].

chebyshev_coefficients(f_values)

Chebyshev T coefficients from values at CGL points.

chebyshev_evaluate(c, x)

Evaluate a Chebyshev expansion via Clenshaw's algorithm.

fourier_points(N, L)

N equally-spaced points on [0, L).