Collection of function spaces.
A function space is composed of elements of basis functions. Each function in the function space can be represented as a linear combination of basis functions, represented by real/complex coefficients (spectral space).
Chebyshev (Orthogonal), see [chebyshev()]ChebDirichlet (Composite), see [cheb_dirichlet()]ChebNeumann (Composite), see [cheb_neumann()]FourierC2c (Orthogonal), see [fourier_c2c()]FourierR2c (Orthogonal), see [fourier_r2c()]A transformation describes a change from physical space to functional space.
For example, a Fourier transform describes a transformation of a function
on a regular grid into coefficients of sine/cosine polynomials.
This is analogous to other function spaces. The transformations are
implemented by the [Transform] trait.
Apply forward transform of 1d array in cheb_dirichlet space
rust
use funspace::{Transform, cheb_dirichlet};
use ndarray::prelude::*;
use ndarray::Array1;
let mut cd = cheb_dirichlet::<f64>(5);
let input = array![1., 2., 3., 4., 5.];
let output: Array1<f64> = cd.forward(&input, 0);
An essential advantage of the representation of a function with coefficients in
the function space is its ease of differentiation. Differentiation in
Fourier space becomes multiplication with the wavenumber vector.
Differentiation in Chebyshev space is done by a recurrence
relation and almost as fast as in Fourier space.
Each base implements a differentiation method, which is applied to
an array of coefficents. This is defined by the [Differentiate] trait.
Apply differentiation
```rust
use funspace::{Transform, Differentiate, Basics, fourierr2c};
use ndarray::prelude::*;
use ndarray::Array1;
use numcomplex::Complex;
// Define base
let mut fo = fourier_r2c(8);
// Get coordinates in physical space
let x = fo.coords().clone();
let v = x.mapv(|xi: f64| (2. * xi).sin());
// Transform to physical space
let vhat: Array1
// Apply differentiation twice along first axis
let dvhat = fo.differentiate(&vhat, 2, 0);
// Transform back to spectral space
let dv: Array1
Bases such as those of Fourier polynomials or Chebyshev polynomials are are considered as orthogonal bases, i.e. the dot product of every single polynomial with any other polynomial in its set vanishes. In these cases the mass matrix is a diagonal matrix. However, other function spaces can be constructed by a linear combination of the orthogonal basis functions. In this way, bases can be constructed construct bases that satisfy certain boundary conditions such as Dirichlet (zero at the ends) or Neumann (zero derivative at the ends). This is useful for solving partial differential equations. If they are expressed in these composite function spaces, the boundary condition is automatically satisfied. This is known as the Galerkin method.
To switch from its composite form to the orthogonal form, each base implements
a [FromOrtho] trait, which defines the transform to_ortho and from_ortho.
If the base is already orthogonal, the input will be returned, otherwise it
is transformed from the composite space to the orthogonal space.
Note that the size of the composite space is usually
less than its orthogonal counterpart. In other words, the composite space is
usually a lower dimensional subspace of the orthogonal space. Therefore the output
array must not maintain the same shape.
Transform composite space cheb_dirichlet to its orthogonal counterpart
chebyshev. Note that cheb_dirichlet has 6 spectral coefficients,
while the chebyshev bases has 8.
```rust
use funspace::{Transform, FromOrtho, Basics};
use funspace::{chebdirichlet, chebyshev};
use std::f64::consts::PI;
use ndarray::prelude::*;
use ndarray::Array1;
use numcomplex::Complex;
// Define base
let mut ch = chebyshev(8);
let mut cd = chebdirichlet(8);
// Get coordinates in physical space
let x = ch.coords().clone();
let v = x.mapv(|xi: f64| (PI / 2. * xi).cos());
// Transform to physical space
let chvhat: Array1
// However, if the physical field values do not
// satisfy dirichlet boundary conditions, they
// will be enforced by the transform to chebdirichle
// and ultimately the transformed values will deviate
// from a pure chebyshev transform (which does not)
// enfore the boundary conditions.
let mut v = x.mapv(|xi: f64| (PI / 2. * xi).sin());
let chvhat: Array1
A collection of bases forms a function space on which one can in turn define operations
along a specfic dimension (= axis). However, to transform a field from the physical space
into the spectral space, special attention must be paid to how the transformations are
concatenated in a multidimensional space. Not all combinations are possible.
For example, cheb_dirichlet is a real-to-real transform,
while fourier_r2c defines a real-to-complex transform.
Thus, for a given real-valued physical field, the Chebyshev transform must precede
the fourier transform in the forward transform, and in reverse order in
the backward transform.
Note: Currently funspace supports 1- 2- and 3 - dimensional spaces.
Apply transform from physical to spectral in a two-dimensional space
rust
use funspace::{fourier_r2c, cheb_dirichlet, Space2, BaseSpace};
use ndarray::prelude::*;
use std::f64::consts::PI;
use num_complex::Complex;
// Define the space and allocate arrays
let mut space = Space2::new(&fourier_r2c(5), &cheb_dirichlet(5));
let mut v: Array2<f64> = space.ndarray_physical();
// Set some field values
let x = space.coords_axis(0);
let y = space.coords_axis(1);
for (i,xi) in x.iter().enumerate() {
for (j,yi) in y.iter().enumerate() {
v[[i,j]] = xi.sin() * (PI/2.*yi).cos();
}
}
// Transform forward (vhat is complex)
let mut vhat = space.forward(&v);
// Transform backward (v is real)
let v = space.backward(&vhat);
An mpi version of funspace can be found on
https://github.com/preiter93/funspace-mpi
The mpi enabled version will be merged
into this crate as soon as rustmpi releases
an updated version on crates.io
License: MIT