The scalar wave equation in Method of Lines form

Ian Hawke

\( \)Date\( \)

Abstract

WaveMoL is an example implementation of a thorn that uses the method of lines thorn MoL. The wave equation in first order form is implemented.

1 Purpose

WaveToy is the simple test thorn that comes with Cactus as standard. This is written so that it solves the wave equation \begin {equation} \label {eq:wave1} \partial _t^2 \phi = \partial _{x^i}^2 \phi ^i \end {equation} directly using the leapfrog scheme. This form of the equations isn’t suitable for use with the method of lines.

The purpose of this thorn is to rewrite the equations in first order form

\begin {eqnarray} \label {eq:wave2} \partial _t \Phi & = & \partial _{x^i} \Pi ^i, \\ \partial _t \Pi ^j & = & \partial _{x^j} \Phi , \\ \partial _t \phi & = & \Phi , \\ \partial _{x^j} \phi & = & \Pi ^j. \end {eqnarray}

The first three equations (which expand to five separate PDEs) will be evolved. The final equation is used to set the initial data and can be thought of as a constraint.

This will be implemented using simple second order differencing in space. Time evolution is performed by the method of lines thorn MoL.

2 How it works

The equations are evolved entirely using the method of lines thorn. So all we have to provide (for the evolution) is a method of calculating the right hand side of equation (??) and boundary conditions. The boundary conditions are standard from wavetoy itself. The right hand side is calculated using second order centred finite differences.

To be compatible with the method of lines thorn we must let it know that the GFs \((\phi , \Pi , \Phi ^j)\) exist and where they are stored. They should all have at least two time levels (although the addition of extra time levels may not cause problems it’s just wasting space). The GFs corresponding to the right hand sides must also be registered with the MoL thorn. These should only have one time level. MoL is informed of the existence of these grid functions through the accumulator parameters and the aliased functions.

3 Parameters


bound	Scope: private	KEYWORD

Description: Type of boundary condition to use

Range		Default: none
none	No boundary condition
flat	Flat boundary condition
radiation	Radiation boundary condition


wavemol_maxnumconstrainedvars	Scope: restricted	INT

Description: The maximum number of constrained variables used by WaveMoL

Range		Default: 1
1:1	The energy


wavemol_maxnumevolvedvars	Scope: restricted	INT

Description: The maximum number of evolved variables used by WaveMoL

Range		Default: 5
5:5	Just 5: phi and the four derivatives


mol_num_constrained_vars	Scope: shared from METHODOFLINES	INT


mol_num_evolved_vars	Scope: shared from METHODOFLINES	INT


mol_num_saveandrestore_vars	Scope: shared from METHODOFLINES	INT

4 Interfaces

General

Implements:

wavemol

Grid Variables

4.0.1 PUBLIC GROUPS


Group Names	Variable Names	Details

scalarevolvemol_scalar		compact	0
	phi	description	The scalar field and time derivative
	phit	dimensions	3
		distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”Scalar”
		timelevels	3
		variable type	REAL

scalarevolvemol_vector		compact	0
	phix	description	The scalar field spatial derivatives
	phiy	dimensions	3
	phiz	distribution	DEFAULT
		group type	GF
		tags	tensortypealias=”U”
		timelevels	3
		variable type	REAL

scalarrhsmol_scalar		compact	0
	phirhs	description	The right hand side for the scalar field
	phitrhs	dimensions	3
		distribution	DEFAULT
		group type	GF
		timelevels	1
		variable type	REAL

scalarrhsmol_vector		compact	0
	phixrhs	description	The right hand side for the scalar field derivatives
	phiyrhs	dimensions	3
	phizrhs	distribution	DEFAULT
		group type	GF
		timelevels	1
		variable type	REAL

energy	energy	compact	0
		description	The energy of the field
		dimensions	3
		distribution	DEFAULT
		group type	GF
		timelevels	1
		variable type	REAL

Uses header:

Symmetry.h

5 Schedule

This section lists all the variables which are assigned storage by thorn CactusExamples/WaveMoL. 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

Always:
scalarevolvemol_scalar[3] scalarevolvemol_vector[3]
scalarrhsmol_scalar scalarrhsmol_vector
energy

Scheduled Functions

CCTK_STARTUP

wavemol_startup

	Language:	c
	Type:	function

CCTK_BASEGRID

wavemol_initsymbound

schedule symmetries

	Language:	c
	Options:	meta
	Type:	function

MoL_Register

wavemol_registervars

	Language:	c
	Options:	meta
	Type:	function

Driver_BoundarySelect

wavemol_selectdriverbc

choose boundary condtion for driver to apply

	Language:	c
	Options:	global
	Type:	function

MoL_CalcRHS

wavemol_calcrhs

	Language:	c
	Reads:	wavemol::phit(everywhere)
		scalarevolvemol_vector
	Type:	function
	Writes:	wavemol::scalarrhsmol_scalar(interior)
		scalarrhsmol_vector

MoL_PostStep

wavemol_boundaries

	Language:	c
	Options:	level
	Sync:	scalarevolvemol_scalar
		scalarevolvemol_vector
	Type:	function

MoL_PostStep

applybcs

apply boundary conditions for wavemol

	After:	wavemol_boundaries
	Type:	group

MoL_PostStep

wavemol_energy

calculate the energy

	After:	wavemol_applybcs
	Language:	c
	Reads:	wavemol::phit(everywhere)
		scalarevolvemol_vector
	Type:	function
	Writes:	wavemol::energy(everywhere)

CCTK_POSTINITIAL

wavemol_energy

calculate the energy

	Language:	c
	Reads:	wavemol::phit(everywhere)
		scalarevolvemol_vector
	Type:	function
	Writes:	wavemol::energy(everywhere)

Aliased Functions

Alias Name:	Function Name:
ApplyBCs	WaveMoL_ApplyBCs