## IDScalarWave

Date

Abstract

Initial Data for the 3D Scalar Wave Equation

### 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

 $\Psi \left(r,t\right)=\frac{1}{r}\left(f\left(r+t\right)+g\left(r-t\right)\right)$ (1)

where the functions $f$ and $g$ can be freely chosen.

Making the additional requirement of time symmetry at $t=0$, forces

 $f\left(r\right)=g\left(r\right)$ (2)

Thus if the solution at t=0 is given by $\varphi \left(r\right)$, the general solution will be

 $\Psi \left(r,t\right)=\frac{1}{2r}\left(\left(r+t\right)\varphi \left(r+t\right)+\left(r-t\right)\varphi \left(r-t\right)\right)$ (3)

### 3 Gaussian

The gaussian solution is spherically symmetric about the origin of the Cartesian coordinate system, and is time symmetric. The initial proﬁle is

 $\varphi \left(r\right)=Aexp\left(-{r}^{2}∕\sigma \right)$ (4)

with the solution at the origin being

 $\Psi \left(r=0,t\right)=\left(1-2\frac{{t}^{2}}{\sigma }\right)exp\left(-{t}^{2}∕\sigma \right)$ (5)

The Gaussian solution is set with the parameters

• amplitude = $A$
• sigma = $\sigma$

### 4 Parameters

 amplitude Scope: private REAL Description: The amplitude of the waves Range Default: 1.0 *:* No restriction

 kx Scope: private REAL Description: The wave number in the x-direction Range Default: 4.0 *:* No restriction

 ky Scope: private REAL Description: The wave number in the y-direction Range Default: 0.0 *:* No restriction

 kz Scope: private REAL Description: The wave number in the z-direction Range Default: 0.0 *:* No restriction

 radius Scope: private REAL Description: The radius of the gaussian wave Range Default: 0.0 0:* Positive

 sigma Scope: private REAL Description: The sigma for the gaussian wave Range Default: 0.1 0:* Positive

 initial_data Scope: restricted KEYWORD Description: Type of initial data Range Default: gaussian plane Plane wave gaussian Gaussian wave box Box wave none No initial data, zero phi

 type Scope: shared from GRID KEYWORD

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.

NONE

#### Scheduled Functions

CCTK_PARAMCHECK

idscalarwave_checkparameters

check parameters

 Language: fortran Type: function

CCTK_INITIAL

wavetoy_initialdata

execute initial data for wavetoy

 Type: group

WaveToy_InitialData

idscalarwave_initialdata

initial data for 3d wave equation

 Language: fortran Storage: wavetoy::scalarevolve[3] Type: function