Semi-infinite spectral mappings¶

The rational and algebraic mappings both transform Chebyshev points on $[-1, 1)$ to the semi-infinite physical interval $[0, \infty)$. This notebook compares their point distributions and the convergence of a mapped first-derivative operator.

In [1]:

import matplotlib.pyplot as plt
import numpy as np

from aerokit.common.mapping import (
    SemiInfiniteAlgebraicMapping,
    SemiInfiniteRationalMapping,
)
from aerokit.common.numspectral import ChebCollocation

L = 1.0
mappings = {
    "Rational": SemiInfiniteRationalMapping(L),
    "Algebraic": SemiInfiniteAlgebraicMapping(L),
}

import matplotlib.pyplot as plt import numpy as np from aerokit.common.mapping import ( SemiInfiniteAlgebraicMapping, SemiInfiniteRationalMapping, ) from aerokit.common.numspectral import ChebCollocation L = 1.0 mappings = { "Rational": SemiInfiniteRationalMapping(L), "Algebraic": SemiInfiniteAlgebraicMapping(L), }

Mapping and collocation-point comparison¶

Both mappings include the origin at $\xi=-1$ and approach infinity at $\xi=1$. The latter endpoint is omitted from the plot.

In [2]:

xi_plot = np.linspace(-1.0, 0.995, 500)
npts = 24
xi_nodes = ChebCollocation(npts).xi

fig, (ax_map, ax_grid) = plt.subplots(1, 2, figsize=(12, 4))
for name, mapping in mappings.items():
    r = mapping.xi_to_x(xi_plot)
    ax_map.plot(xi_plot, r, label=name)
    r_nodes = mapping.xi_to_x(xi_nodes)
    finite = np.isfinite(r_nodes)
    ax_grid.plot(r_nodes[finite], np.zeros(finite.sum()), "o", label=name)

ax_map.set(xlabel=r"$\xi$", ylabel=r"$r/L$", ylim=(0, 20), title="Coordinate maps")
ax_map.legend()
ax_grid.set(xlabel=r"$r/L$", title=f"Finite collocation points ($N={npts}$)", yticks=[])
ax_grid.legend()
fig.tight_layout()

xi_plot = np.linspace(-1.0, 0.995, 500) npts = 24 xi_nodes = ChebCollocation(npts).xi fig, (ax_map, ax_grid) = plt.subplots(1, 2, figsize=(12, 4)) for name, mapping in mappings.items(): r = mapping.xi_to_x(xi_plot) ax_map.plot(xi_plot, r, label=name) r_nodes = mapping.xi_to_x(xi_nodes) finite = np.isfinite(r_nodes) ax_grid.plot(r_nodes[finite], np.zeros(finite.sum()), "o", label=name) ax_map.set(xlabel=r"$\xi$", ylabel=r"$r/L$", ylim=(0, 20), title="Coordinate maps") ax_map.legend() ax_grid.set(xlabel=r"$r/L$", title=f"Finite collocation points ($N={npts}$)", yticks=[]) ax_grid.legend() fig.tight_layout()

First-derivative convergence¶

For $f(r)=\exp(-r^2)$, the exact derivative is $f'(r)=-2r\exp(-r^2)$. ChebCollocation creates only reference-coordinate matrices; each mapping applies the chain rule to obtain the physical derivative matrix.

In [3]:

def derivative_error(npts, mapping):
    reference = ChebCollocation(npts)
    reference.compute_matder(1)
    derivative = mapping.transform_derivative_matrices(reference._matder)[:, :, 0]

    r = mapping.xi_to_x(reference.xi)
    values = np.exp(-r**2)
    exact = np.zeros_like(r)
    finite = np.isfinite(r)
    exact[finite] = -2.0 * r[finite] * values[finite]
    return np.sqrt(np.mean((derivative @ values - exact) ** 2))

sizes = np.array([12, 16, 20, 28, 36, 48, 64])
fig, ax = plt.subplots(figsize=(6, 4))
for name, mapping in mappings.items():
    errors = [derivative_error(npts, mapping) for npts in sizes]
    ax.semilogy(sizes, errors, "o-", label=name)

ax.set(xlabel="Number of collocation points", ylabel="RMS error in $f'(r)$", title="Mapped first-derivative convergence")
ax.grid(True, which="both")
ax.legend();

def derivative_error(npts, mapping): reference = ChebCollocation(npts) reference.compute_matder(1) derivative = mapping.transform_derivative_matrices(reference._matder)[:, :, 0] r = mapping.xi_to_x(reference.xi) values = np.exp(-r**2) exact = np.zeros_like(r) finite = np.isfinite(r) exact[finite] = -2.0 * r[finite] * values[finite] return np.sqrt(np.mean((derivative @ values - exact) ** 2)) sizes = np.array([12, 16, 20, 28, 36, 48, 64]) fig, ax = plt.subplots(figsize=(6, 4)) for name, mapping in mappings.items(): errors = [derivative_error(npts, mapping) for npts in sizes] ax.semilogy(sizes, errors, "o-", label=name) ax.set(xlabel="Number of collocation points", ylabel="RMS error in $f'(r)$", title="Mapped first-derivative convergence") ax.grid(True, which="both") ax.legend();