Time

Gabrielle Allen

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.
$i Δt = dtfac ∗min(Δx )$
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
$√---- Δt = courantfac ∗min(Δxi)∕courantwavespeed∕ dim$
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
$√ ---- Δt = courantfac∗courantmintime∕ dim$
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

√---- Δt ≤ min (Δxi )∕wave speed∕ dim

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

General

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