A 3D spherical wave solution to the Euclidean wave equation that is regular at the origin. More...

#include <RegularSphericalWave.hpp>

Classes
struct	Profile

Public Types
using	options = tmpl::list<Profile>
using	tags

Public Member Functions
	RegularSphericalWave (std::unique_ptr< MathFunction< 1, Frame::Inertial > > profile)
	RegularSphericalWave (const RegularSphericalWave &other)
RegularSphericalWave &	operator= (const RegularSphericalWave &other)
	RegularSphericalWave (RegularSphericalWave &&)=default
RegularSphericalWave &	operator= (RegularSphericalWave &&)=default
auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override
tuples::TaggedTuple< Tags::Psi, Tags::Pi, Tags::Phi< 3 > >	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list< Tags::Psi, Tags::Pi, Tags::Phi< 3 > >) const
tuples::TaggedTuple<::Tags::dt< Tags::Psi >, ::Tags::dt< Tags::Pi >, ::Tags::dt< Tags::Phi< 3 > > >	variables (const tnsr::I< DataVector, 3 > &x, double t, tmpl::list<::Tags::dt< Tags::Psi >, ::Tags::dt< Tags::Pi >, ::Tags::dt< Tags::Phi< 3 > > >) const
void	pup (PUP::er &p) override

Static Public Attributes
static constexpr size_t	volume_dim = 3
static constexpr Options::String	help

Friends
bool	operator== (const RegularSphericalWave &lhs, const RegularSphericalWave &rhs)
bool	operator!= (const RegularSphericalWave &lhs, const RegularSphericalWave &rhs)

Detailed Description

A 3D spherical wave solution to the Euclidean wave equation that is regular at the origin.

The solution is given by \(\Psi(\vec{x},t) = \Psi(r,t) = \frac{F(r-t)-F(-r-t)}{r}\) describing an outgoing and an ingoing wave with profile \(F(u)\). For small \(r\) the solution is approximated by its Taylor expansion \(\Psi(r,t)=2 F^\prime(-t) + \mathcal{O}(r^2)\). The outgoing and ingoing waves meet at the origin (and cancel each other) when \(F^\prime(-t)=0\).

The expansion is employed where \(r\) lies within the cubic root of the machine epsilon. Inside this radius we expect the error due to the truncation of the Taylor expansion to be smaller than the numerical error made when evaluating the full \(\Psi(r,t)\). This is because the truncation error scales as \(r^2\) (since we keep the zeroth order, and the linear order vanishes as all odd orders do) and the numerical error scales as \(\frac{\epsilon}{r}\), so they are comparable at \(r\propto\epsilon^\frac{1}{3}\).

Requires: the profile \(F(u)\) to have a length scale of order unity so that "small" \(r\) means \(r\ll 1\). This is without loss of generality because of the scale invariance of the wave equation. The profile could be a Gausssian centered at 0 with width 1, for instance.

Member Typedef Documentation

◆ tags

using ScalarWave::Solutions::RegularSphericalWave::tags

Initial value:

 
      tmpl::list<Tags::Psi, Tags::Pi, Tags::Phi<3>, ::Tags::dt<Tags::Psi>,
                 ::Tags::dt<Tags::Pi>, ::Tags::dt<Tags::Phi<3>>>

Member Function Documentation

◆ get_clone()

auto ScalarWave::Solutions::RegularSphericalWave::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

◆ help

Options::String ScalarWave::Solutions::RegularSphericalWave::help

staticconstexpr

Initial value:

= {
      "A spherical wave solution of the Euclidean wave equation that is "
      "regular at the origin"}

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

src/PointwiseFunctions/AnalyticSolutions/WaveEquation/RegularSphericalWave.hpp

Classes

Public Types

Public Member Functions

Static Public Attributes

Friends