\( f(i) = \begin{cases} 1 \text{ if } \eta < \eta_c \\ \exp\left( -\alpha \left(\frac{\eta-\eta_c}{1-\eta_c} \right)^{2s}\right) \text { if } \eta \geq \eta_c \\ 0 \text{ else} \\ \eta=\frac{i}{1-n} \end{cases}\) More...

Public Member Functions
	ExponentialFilter (double alpha, double eta_c, unsigned order, unsigned n)
	Create exponential filter \( \begin{cases} 1 \text{ if } \eta < \eta_c \\ \exp\left( -\alpha \left(\frac{\eta-\eta_c}{1-\eta_c} \right)^{2s}\right) \text { if } \eta \geq \eta_c \\ 0 \text{ else} \\ \eta := \frac{i}{n-1} \end{cases}\). More...

double	operator() (unsigned i) const

Detailed Description

\( f(i) = \begin{cases} 1 \text{ if } \eta < \eta_c \\ \exp\left( -\alpha \left(\frac{\eta-\eta_c}{1-\eta_c} \right)^{2s}\right) \text { if } \eta \geq \eta_c \\ 0 \text{ else} \\ \eta=\frac{i}{1-n} \end{cases}\)

where n is the number of polynomial coefficients

This function is s times continuously differentiable everywhere

See also: Its main use comes from the application in dg::ModalFilter

Constructor & Destructor Documentation

◆ ExponentialFilter()

dg::ExponentialFilter::ExponentialFilter	(	double	alpha,
		double	eta_c,
		unsigned	order,
		unsigned	n
	)

inline

Create exponential filter \( \begin{cases} 1 \text{ if } \eta < \eta_c \\ \exp\left( -\alpha \left(\frac{\eta-\eta_c}{1-\eta_c} \right)^{2s}\right) \text { if } \eta \geq \eta_c \\ 0 \text{ else} \\ \eta := \frac{i}{n-1} \end{cases}\).

Parameters

alpha	damping for the highest mode is `exp`( -alpha)
eta_c	cutoff frequency (0<eta_c<1), 0.5 or 0 are good starting values
order	8 or 16 are good values
n	The number of polynomial coefficients

Member Function Documentation

◆ operator()()

double dg::ExponentialFilter::operator() ( unsigned i ) const