SpECTRE  v2021.12.06
domain::CoordinateMaps::Interval Class Reference

Maps \(\xi\) in the 1D interval \([A, B]\) to \(x\) in the interval \([a, b]\) according to a domain::CoordinateMaps::Distribution. More...

#include <Interval.hpp>

Public Member Functions

 Interval (double A, double B, double a, double b, Distribution distribution, std::optional< double > singularity_pos=std::nullopt)
 
template<typename T >
std::array< tt::remove_cvref_wrap_t< T >, 1 > operator() (const std::array< T, 1 > &source_coords) const
 
std::optional< std::array< double, 1 > > inverse (const std::array< double, 1 > &target_coords) const
 
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 1, Frame::NoFramejacobian (const std::array< T, 1 > &source_coords) const
 
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 1, Frame::NoFrameinv_jacobian (const std::array< T, 1 > &source_coords) const
 
void pup (PUP::er &p)
 
bool is_identity () const
 

Static Public Attributes

static constexpr size_t dim = 1
 

Friends

bool operator== (const Interval &lhs, const Interval &rhs)
 

Detailed Description

Maps \(\xi\) in the 1D interval \([A, B]\) to \(x\) in the interval \([a, b]\) according to a domain::CoordinateMaps::Distribution.

Details

The mapping takes a domain::CoordinateMaps::Distribution and distributes the grid points accordingly.

The formula for the mapping is, in case of a Linear distribution

\begin{align} x &= \frac{b}{B-A} (\xi-A) + \frac{a}{B-A} (B-\xi)\\ \xi &=\frac{B}{b-a} (x-a) + \frac{A}{b-a} (b-x) \end{align}

For every other distribution we use this linear mapping to map onto an interval \([-1, 1]\), so we define:

\begin{align} f(\xi) &:= \frac{A+B-2\xi}{A-B} \in [-1, 1]\\ g(x) &:= \frac{a+b-2x}{a-b} \in [-1, 1] \end{align}

With this an Equiangular distribution is described by

\begin{align} x &= \frac{a}{2} \left(1-\mathrm{tan}\left(\frac{\pi}{4}f(\xi)\right)\right) + \frac{b}{2} \left(1+\mathrm{tan}\left(\frac{\pi}{4}f(\xi)\right)\right)\\ \xi &= \frac{A}{2} \left(1-\frac{4}{\pi}\mathrm{arctan}\left(g(x)\right)\right) + \frac{B}{2} \left(1+\frac{4}{\pi}\mathrm{arctan}\left(g(x)\right)\right) \end{align}

Note
The equiangular distribution is intended to be used with the Wedge map when equiangular coordinates are chosen for those maps. For more information on this choice of coordinates, see the documentation for Wedge.

For both the Logarithmic and Inverse distribution, we first specify a position for the singularity \(c:=\)singularity_pos outside the target interval \([a, b]\).

The Logarithmic distribution further requires the introduction of a variable \(\sigma\) dependent on whether \(c\) is left ( \( \sigma = 1\)) or right ( \( \sigma = -1\)) of the target interval. With this, the Logarithmic distribution is described by

\begin{align} x &= \sigma\, {\rm exp}\left(\frac{\ln(b-c)+\ln(a-c)}{2} + f(\xi) \frac{\ln(b-c)-\ln(a-c)}{2}\right) + c\\ \xi &= \frac{B-A}{2}\frac{2\ln(\sigma [x-c]) - \ln(b-c) - \ln(a-c)}{\ln(b-c) - \ln(a-c)} + \frac{B+A}{2} \end{align}

and the Inverse distribution by

\begin{align} x &= \frac{2(a-c)(b-c)}{a+b-2c+(a-b)f(\xi)} + c\\ \xi &= -\frac{A-B}{2(a-b)} \left(\frac{2(a-c)(b-c)}{x-c} - (a-c) - (b-c)\right) + \frac{A+B}{2} \end{align}


The documentation for this class was generated from the following file: