1 Introduction

Thorn TmunuBase provides core infrastructure for thorns implementing some kind of energy or matter in general relativity, for example general relativistic hydrodynamics formulations. It provides the basic variables, i.e., the stress-energy tensor \(T_{\mu \nu }\), in addition to a set of parameters to regulate their use. These variables are used to communicate between (possibly multiple) thorns contributing to the stress-energy content of the spacetime, and thorns needing to evaluate the stress-energy tensor such as spacetime evolution methods. It also provides schedule groups to manage when \(T_{\mu \nu }\) is calculated and when it is ready for access.

2 Using TmunuBase

2.1 Variables

TmunuBase weakly assumes (but does not require) that the spacetime is described in terms of a \(3+1\) decomposition. The variables provided by TmunuBase are:

• The “scalar” part of \(T_{\mu \nu }\), its time-time component: eTtt

• The “vector” part of \(T_{\mu \nu }\), its time-space components: eTtx, eTty, eTtz

• The “tensor” part of \(T_{\mu \nu }\), its space-space components: eTxx, eTxy, eTxz, eTyy, eTyz, eTzz

These components have the prefix e to avoid naming conflicts with existing variables. Many thorns dealing with matter already use variable names such as Ttt.

These variables have up to three time levels.

2.2 Parameters

By default, the TmunuBase variables have no storage, and TmunuBase is inactive. This makes it possible to add a matter interface to existing vacuum spacetime methods without changing their behaviour.

Several parameters choose how TmunuBase behaves at run time:

• The parameter stress_energy_storage activates storage for \(T_{\mu \nu }\) and enables the schedule groups which calculate it.

• The parameter stress_energy_at_RHS moves calculating the \(T_{\mu \nu }\) from the evol bin into the MoL_PostStep group. This increases the order of accuracy of the spacetime–matter coupling, but is only possible when thorn MoL is used.¹ Generally, this parameter should be set when MoL is used.

• The parameter timelevels chooses the number of time levels for \(T_{\mu \nu }\). The default is a single time level, which is sufficient for unigrid simulation. Mesh refinement simulation may require several time levels if mesh refinement boundaries require correct values.

• The parameter prolongation_type defines the prolongation operator for mesh refinement boundaries. The default is Lagrange interpolation.

The grid scalar stress_energy_state describes whether the \(T_{\mu \nu }\) variables have storage.

3 Programming with TmunuBase

3.1 Contributing to \(T_{\mu \nu }\)

There may be multiple thorns contributing to \(T_{\mu \nu }\). Therefore, thorn TmunuBase initialises \(T_{\mu \nu }\) to zero, and each thorn has to add to the existing values in \(T_{\mu \nu }\). The corresponding routine should be scheduled in the bin AddToTmunu. Note: Do not schedule anything in the schedule bin SetTmunu.

3.2 Reading from \(T_{\mu \nu }\)

Since the values of \(T_{\mu \nu }\) change at each time step, or – if a thorn like MoL is used – at each substep, \(T_{\mu \nu }\) needs to be recalculated frequently. This happens either in the schedule bin evol or in the schedule group MoL_PostStep. \(T_{\mu \nu }\) may only be accessed after it has been calculated, e.g. IN MoL_PostStep AFTER SetTmunu. \(T_{\mu \nu }\) can be freely accessed at other times, e.g. in MoL_CalcRHS or at poststep or analyisis.

3.3 Acknowledgements

We thank I. Hawke for designing and implementing thorn MoL, without which a generic high-order coupling between spacetime and hydrodynamics methods would not be possible.

4 Parameters

prolongation_type	Scope: private	STRING
Description: The kind of boundary prolongation for the stress-energy tensor
Range		Default: Lagrange
agrange$	standard prolongation (requires several time levels)
one$	no prolongation (use this if you do not have enough time levels active)
	any other supported prolongation type

5 Interfaces

General

Implements:

tmunubase

Inherits:

admbase

staticconformal

Grid Variables

5.0.1 PUBLIC GROUPS

6 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinBase/TmunuBase. 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

Scheduled Functions

CCTK_WRAGH

  tmunubase_setstressenergystate

  set the stress_energy_state variable

CCTK_PARAMCHECK

  tmunubase_paramcheck

  check that no deprecated parameters are used.

MoL_PostStep (conditional)

  settmunu

  group for calculating the stress-energy tensor

SetTmunu (conditional)

  tmunubase_zerotmunu

  initialise the stress-energy tensor to zero

SetTmunu (conditional)

  addtotmunu

  add to the stress-energy tensor here