The scalar wave equation in Method of Lines form

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

 ${\partial }_{t}^{2}\varphi ={\partial }_{{x}^{i}}^{2}{\varphi }^{i}$ (1)

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

$\begin{array}{rcll}{\partial }_{t}\Phi & =& {\partial }_{{x}^{i}}{\Pi }^{i},& \text{(2)}\text{}\text{}\\ {\partial }_{t}{\Pi }^{j}& =& {\partial }_{{x}^{j}}\Phi ,& \text{(3)}\text{}\text{}\\ {\partial }_{t}\varphi & =& \Phi ,& \text{(4)}\text{}\text{}\\ {\partial }_{{x}^{j}}\varphi & =& {\Pi }^{j}.& \text{(5)}\text{}\text{}\end{array}$

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 (2) 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 $\left(\varphi ,\Pi ,{\Phi }^{j}\right)$ 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

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

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

register banner

 Language: c Type: function

CCTK_BASEGRID

wavemol_initsymbound

schedule symmetries

 Language: c Options: meta Type: function

MoL_Register

wavemol_registervars

register variables for mol

 Language: c Options: meta Type: function

MoL_CalcRHS

wavemol_calcrhs

register rhs calculation for mol

 Language: c Type: function

MoL_PostStep

wavemol_energy

calculate the energy

 Language: c Type: function

CCTK_POSTINITIAL

wavemol_energy

calculate the energy

 Language: c Type: function

MoL_PostStep

wavemol_boundaries

register boundary enforcement in mol

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

MoL_PostStep

applybcs

apply boundary conditions for wavemol

 After: wavemol_boundaries Type: group

Aliased Functions

 Alias Name: Function Name: ApplyBCs WaveMoL_ApplyBCs