1 Purpose

This thorn provides different initial data for the 3D scalar wave equation.

2 Spherically Symmetric Solutions

The general spherically symmetric solution can be written \begin {equation} \Psi (r,t) = \frac {1}{r}\left (f(r+t)+g(r-t)\right ) \end {equation} where the functions \(f\) and \(g\) can be freely chosen.

Making the additional requirement of time symmetry at \(t=0\), forces \begin {equation} f(r)=g(r) \end {equation} Thus if the solution at t=0 is given by \(\phi (r)\), the general solution will be \begin {equation} \Psi (r,t) = \frac {1}{2r}\left ( (r+t)\phi (r+t)+(r-t)\phi (r-t) \right ) \end {equation}

3 Gaussian

The gaussian solution is spherically symmetric about the origin of the Cartesian coordinate system, and is time symmetric. The initial profile is \begin {equation} \phi (r) = A \exp (- r^2/\sigma ) \end {equation} with the solution at the origin being \begin {equation} \Psi (r=0,t) = \left (1-2\frac {t^2}{\sigma }\right )\exp (-t^2/\sigma ) \end {equation}

The Gaussian solution is set with the parameters

• amplitude = \(A\)

• sigma = \(\sigma \)

4 Parameters

5 Interfaces

General

Implements:

idscalarwave

Inherits:

wavetoy

grid

6 Schedule

This section lists all the variables which are assigned storage by thorn CactusWave/IDScalarWave. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

NONE

Scheduled Functions

CCTK_PARAMCHECK

  idscalarwave_checkparameters

  check parameters

CCTK_INITIAL

  wavetoy_initialdata

  execute initial data for wavetoy

WaveToy_InitialData

  idscalarwave_initialdata

  initial data for 3d wave equation