Grav-Sim

 

Accuracy of the Grav-Sim Integrators

The graphs in this article were generated by simulating the same Grav-Sim 2-body model for 201 orbits and drawing every 20th orbit.

The following initial conditions were used an should lead to successive elliptical orbits of eccenticity 0.78:

For each integrator, graphs are shown for levels 1, 2 and 3 for both Constant and Variable time-steps.

The settings for each integrator have been manually tuned to use as many steps per orbit as necessary to produce graphs that appear similar at level 1 (i.e. have approximately the same magnitude of accumulated errors after 201 orbits). This is necessarily subjective.

Each successive level uses a factor of square-root-of-2 times more steps than the previous level. Hence level 3 uses twice as many steps as level 1.

The accuracy at levels 2, 3 and beyond depends on the quality of the integrator itself - in particular the "order" - which dictates how quickly the errors reduce with increasing numbers of steps.

Forward Euler (1st Order)

Level	Constant Time-Steps	Variable Time-Steps
1	500000 steps per orbit	144000 steps per orbit
2	707106 steps per orbit	203646 steps per orbit
3	1000000 steps per orbit	288000 steps per orbit

Forward Plus (1st Order)

Level	Constant Time-Steps	Variable Time-Steps
1	250000 steps per orbit	72000 steps per orbit
2	353554 steps per orbit	101823 steps per orbit
3	1000000 steps per orbit	144000 steps per orbit

ProtoHermite (2nd Order)

Level	Constant Time-Steps	Variable Time-Steps
1	900 steps per orbit	400 steps per orbit
2	1273 steps per orbit	566 steps per orbit
3	1800 steps per orbit	800 steps per orbit

RungeKutta2n (2nd Order)

Level	Constant Time-Steps	Variable Time-Steps
1	8000 steps per orbit	1151 steps per orbit
2	11313 steps per orbit	1628 steps per orbit
3	16000 steps per orbit	2302 steps per orbit

RungeKutta2n_fsal (2nd Order)

Level	Constant Time-Steps	Variable Time-Steps
1	2000 steps per orbit	287 steps per orbit
2	2828 steps per orbit	406 steps per orbit
3	4000 steps per orbit	574 steps per orbit

LeapFrog (2nd Order)

Level	Constant Time-Steps	Variable Time-Steps
1	500 steps per orbit	143 steps per orbit
2	707 steps per orbit	202 steps per orbit
3	1000 steps per orbit	286 steps per orbit

RungeKutta3 (3rd Order)

Level	Constant Time-Steps	Variable Time-Steps
1	1000 steps per orbit	333 steps per orbit
2	1414 steps per orbit	471 steps per orbit
3	2000 steps per orbit	666 steps per orbit

RungeKutta4 (4th Order)

Level	Constant Time-Steps	Variable Time-Steps
1	666 steps per orbit	216 steps per orbit
2	942 steps per orbit	305 steps per orbit
3	1332 steps per orbit	432 steps per orbit

RungeKutta4n (4th Order)

Level	Constant Time-Steps	Variable Time-Steps
1	666 steps per orbit	216 steps per orbit
2	942 steps per orbit	305 steps per orbit
3	1332 steps per orbit	432 steps per orbit

ChinChen (4th Order)

Level	Constant Time-Steps	Variable Time-Steps
1	666 steps per orbit	216 steps per orbit
2	942 steps per orbit	305 steps per orbit
3	1332 steps per orbit	432 steps per orbit

ChinChenOptimised (4th Order)

Level	Constant Time-Steps	Variable Time-Steps
1	666 steps per orbit	216 steps per orbit
2	942 steps per orbit	305 steps per orbit
3	1332 steps per orbit	432 steps per orbit

Hermite (4th Order)

Level	Constant Time-Steps	Variable Time-Steps
1	250 steps per orbit	75 steps per orbit
2	353 steps per orbit	106 steps per orbit
3	500 steps per orbit	150 steps per orbit

Hermite6 (6th Order)

Level	Constant Time-Steps	Variable Time-Steps
1	193 steps per orbit	58 steps per orbit
2	273 steps per orbit	82 steps per orbit
3	386 steps per orbit	116 steps per orbit

Hermite8 (8th Order)

Level	Constant Time-Steps	Variable Time-Steps
1	148 steps per orbit	45 steps per orbit
2	209 steps per orbit	64 steps per orbit
3	296 steps per orbit	90 steps per orbit

Note that the number of steps taken is an approximate target and will vary a little in practice, due to:

• The shape of the orbit (e.g. circular vs elongated)
• The accumulation of errors

There are 2 tuning variables in the Grav-Sim source code that control the relationship between the number of steps per orbit and the collision timescale (which is the only statistic available at run time). The settings for these variables is also subjective. For reference, they appear in basic.h and were set to the following values when generating these graphs:

#define CONSTANT_TUNING_FACTOR 5.6

#define VARIABLE_TUNING_FACTOR 18.0