## Time

Date

### Abstract

Calculates the timestep used for an evolution

### 1 Purpose

This thorn provides routines for calculating the timestep for an evolution based on the spatial Cartesian grid spacing and a wave speed.

### 2 Description

Thorn Time uses one of four methods to decide on the timestep to be used for the simulation. The method is chosen using the keyword parameter time::timestep_method.

• time::timestep_method = "given"

The timestep is fixed to the value of the parameter time::timestep.

• time::timestep_method = "courant_static"

This is the default method, which calculates the timestep once at the start of the simulation, based on a simple courant type condition using the spatial gridsizes and the parameter time::dtfac.

$\mathrm{\Delta }t=\text{dtfac}\ast \text{min}\left(\mathrm{\Delta }{x}^{i}\right)$

Note that it is up to the user to custom dtfac to take into account the dimension of the space being used, and the wave speed.

• time::timestep_method = "courant_speed"

This choice implements a dynamic courant type condition, the timestep being set before each iteration using the spatial dimension of the grid, the spatial grid sizes, the parameter courant_fac and the grid variable courant_wave_speed. The algorithm used is

$\mathrm{\Delta }t=\text{courant_fac}\ast \text{min}\left(\mathrm{\Delta }{x}^{i}\right)∕\text{courant_wave_speed}∕\sqrt{\text{d}im}$

For this algorithm to be successful, the variable courant_wave_speed must have been set by some thorn to the maximum propagation speed on the grid before this thorn sets the timestep, that is AT POSTSTEP BEFORE Time_Courant (or earlier in the evolution loop). [Note: The name courant_wave_speed was poorly chosen here, the required speed is the maximum propagation speed on the grid which may be larger than the maximum wave speed (for example with a shock wave in hydrodynamics, also it is possible to have propagation without waves as with a pure advection equation).

• time::timestep_method = "courant_time"

This choice is similar to the method courant_speed above, in implementing a dynamic timestep. However the timestep is chosen using

$\mathrm{\Delta }t=\text{courant_fac}\ast \text{courant_min_time}∕\sqrt{\text{d}im}$

where the grid variable courant_min_time must be set by some thorn to the minimum time for a wave to cross a gridzone before this thorn sets the timestep, that is AT POSTSTEP BEFORE Time_Courant (or earlier in the evolution loop).

In all cases, Thorn Time sets the Cactus variable cctk_delta_time which is passed as part of the macro CCTK_ARGUMENTS to thorns called by the scheduler.

Note that for hyperbolic problems, the Courant condition gives a minimum requirement for stability, namely that the numerical domain of dependency must encompass the physical domain of dependency, or

### 3 Examples

Fixed Value Timestep

time::timestep_method = "given"
time::timestep        = 0.1


Calculate Static Timestep Based on Grid Spacings

The following parameters set the timestep to be 0.25

grid::dx    = 0.5
grid::dy    = 1.0
grid::dz    = 1.0
time::timestep_method = "courant_static"
time::dtfac = 0.5


### 4 Parameters

 courant_fac Scope: private REAL Description: The courant timestep condition dt = courant_fac*max(delta_space)/speed/sqrt(dim) Range Default: 0.9 0:* For positive timestep *:0 For negative timestep

 dtfac Scope: private REAL Description: The standard timestep condition dt = dtfac*max(delta_space) Range Default: 0.5 0:* For positive timestep *:0 For negative timestep

 timestep Scope: private REAL Description: Absolute value for timestep Range Default: 0.0 *:* Could be anything

 timestep_outevery Scope: private INT Description: How often to output courant timestep Range Default: 1 1:* Zero means no output

 verbose Scope: private BOOLEAN Description: Give selective information about timestep setting Default: no

 timestep_method Scope: restricted KEYWORD Description: Method for calculating timestep Range Default: courant_static given Use given timestep courant_static Courant condition at BASEGRID (using dtfac) courant_speed Courant condition at POSTSTEP (using wavespeed and courant_fac) courant_time Courant condition at POSTSTEP (using min time and courant_fac)

 timestep_outonly Scope: restricted BOOLEAN Description: Don’t set a dynamic timestep, just output what it would be Default: no

 cctk_final_time Scope: shared from CACTUS REAL

 terminate Scope: shared from CACTUS KEYWORD

### 5 Interfaces

Implements:

time

#### Grid Variables

##### 5.0.1 PRIVATE GROUPS
 Group Names Variable Names Details couranttemps compact 0 courant_dt description Variable just for output dimensions 0 distribution CONSTANT group type SCALAR timelevels 1 variable type REAL

##### 5.0.2 PUBLIC GROUPS
 Group Names Variable Names Details speedvars compact 0 courant_wave_speed description Speed to use for Courant condition courant_min_time dimensions 0 distribution CONSTANT group type SCALAR timelevels 1 variable type REAL

### 6 Schedule

This section lists all the variables which are assigned storage by thorn CactusBase/Time. 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: speedvars couranttemps

#### Scheduled Functions

CCTK_BASEGRID

time_initialise

initialise time variables

 Before: time_simple Language: c Options: global Type: function Writes: time::speedvars time::courant_dt(everywhere)

CCTK_BASEGRID (conditional)

time_simple

set timestep based on courant condition (courant_static)

 After: spatialspacings Language: c Options: singlemap Type: function

CCTK_BASEGRID (conditional)

time_courant

set timestep based on courant condition (courant_speed)

 After: spatialspacings Language: c Options: singlemap Reads: time::speedvars Type: function Writes: time::courant_dt(everywhere)

CCTK_POSTSTEP (conditional)

time_courant

reset timestep each iteration

 After: spatialspacings Language: c Options: singlemap Reads: time::speedvars Type: function Writes: time::courant_dt(everywhere)

CCTK_BASEGRID (conditional)

time_simple

set timestep based on courant condition (courant_time)

 After: spatialspacings Language: c Options: singlemap Type: function

CCTK_POSTSTEP (conditional)

time_courant

reset timestep each iteration

 After: spatialspacings Language: c Options: singlemap Reads: time::speedvars Type: function Writes: time::courant_dt(everywhere)

CCTK_BASEGRID (conditional)

time_given

set fixed timestep

 After: spatialspacings Language: c Options: singlemap Type: function

#### Aliased Functions

 Alias Name: Function Name: Time_Courant TemporalSpacings Time_Given TemporalSpacings Time_Simple TemporalSpacings