Spectral::filtering Namespace Reference

Matrices for filtering spectral coefficients. More...

Functions
Matrix	exponential_filter (const Mesh< 1 > &mesh, double alpha, unsigned half_power)
	Returns a Matrix by which to multiply the nodal coefficients to apply a stable exponential filter. More...
const Matrix &	zero_lowest_modes (const Mesh< 1 > &mesh, size_t number_of_modes_to_zero)
	Zeros the lowest number_of_modes_to_zero modal coefficients. Note that the matrix must be applied to a nodal representation. More...

Detailed Description

Matrices for filtering spectral coefficients.

Function Documentation

exponential_filter()

Matrix Spectral::filtering::exponential_filter	(	const Mesh< 1 > &	mesh,
		double	alpha,
		unsigned	half_power
	)

Returns a Matrix by which to multiply the nodal coefficients to apply a stable exponential filter.

The exponential filter rescales the modal coefficients according to:

$$\begin{array}{r}{c}_i\to c_i\exp [-\alpha {\left(\frac{i}{N}\right)}^{2m}]\end{array}$$

where $c_i$ are the zero-indexed modal coefficients, $N$ is the basis degree (number of grid points per element per dimension minus one), $\alpha$ determines how much the coefficients are rescaled, and $m$ determines how aggressive/broad the filter is (lower values means filtering more coefficients). Setting $\alpha =36$ results in setting the highest coefficient to machine precision, effectively zeroing it out.

Note

The filter matrix is not cached by the function because it depends on a double, an integer, and the mesh, which could make caching very memory intensive. The caller of this function is responsible for determining whether or not the matrix should be cached.

zero_lowest_modes()

const Matrix & Spectral::filtering::zero_lowest_modes	(	const Mesh< 1 > &	mesh,
		size_t	number_of_modes_to_zero
	)

Zeros the lowest number_of_modes_to_zero modal coefficients. Note that the matrix must be applied to a nodal representation.

Given a function $u$

$$\begin{array}{}\text{(1)}& u(x)=\sum _{i=0}^N{c}_i{P}_i(x),\end{array}$$

where $c_i$ are the modal coefficients and $P_i(x)$ is the basis (e.g. Legendre polynomials), the filter matrix will take the nodal representation of $u$ and zero out the lowest number_of_modes_to_zero modal coefficients. That is, after the filter is applied $u\to \overline{u}$ is

$$\begin{array}{}\text{(2)}& \overline{u}(x)=\sum _{i=k}^N{c}_i{P}_i(x),\end{array}$$

where $k$ is the number of modes set to zero. The output $\overline{u}$ is also in the nodal representation.