Collection of function spaces.
Collection of function spaces.
Chebyshev (Orthogonal), see [chebyshev()]ChebDirichlet (Composite), see [cheb_dirichlet()]ChebNeumann (Composite), see [cheb_neumann()]ChebDirichletNeumann (Composite), see [cheb_dirichlet_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 transforms a function
on a equispaced grid into coefficients of sine/cosine polynomials.
This is analogous to other function spaces. The transformations are
implemented by the [FunspaceElemental] trait.
Apply forward transform of 1d array in cheb_dirichlet space
rust
use funspace::traits::FunspaceElemental;
use funspace::cheb_dirichlet;
use ndarray::{Array1, array};
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 [FunspaceElemental] trait.
Apply differentiation
```rust
use funspace::fourierr2c;
use funspace::traits::{FunspaceElemental, FunspaceExtended};
use ndarray::Array1;
use numcomplex::Complex;
// Define base
let mut fo = fourierr2c(8);
// Get coordinates in physical space
let x: Vec
// Apply differentiation twice along first axis
let dvhat = fo.differentiate(&vhat, 2, 0);
// Transform back to spectral space
let dv: Array1
Function spaces such as those of Fourier polynomials or Chebyshev polynomials are are considered orthogonal, i.e. the dot product of every single polynomial with any other polynomial in its set vanishes. In this cases the mass matrix is a purely 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 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 (see Galerkin method), since calculation in these bases automatically satisfy the boundary conditions.
To switch from its composite form to the orthogonal form, each base implements
a [FunspaceElemental] 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 (to_ortho), or vice versa
(from_ortho).
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 number of coefficients
between orthogonal and composite space is different.
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::traits::{FunspaceElemental, FunspaceExtended};
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: Vec
// 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: Array1
Funspace comes with limited mpi support. Currently this is restricted
to 2D spaces. Under the hood it uses a fork of the rust mpi libary
https://github.com/rsmpi/rsmpi which requires an existing MPI implementation
and libclang.
Activate the feature in your `Cargo.toml
funspace = {version = "0.3", features = ["mpi"]}`
examples/space_mpi.rs
Install cargo mpirun, for example, and run
rust
cargo mpirun --np 2 --example space_mpi --features="mpi"
License: MIT