The Integration Methods in nMod: Part 3

The Runge-Kutta Fourth Order Integration Method

The Runge-Kutta Fourth Order (RK4) Integration function is the most accurate integration method provided in nMod's n-Body model. This accuracy does come at the price of an additional computational burden when compared to the simpler Midpoint method, but it is better suited for experiments where accuracy is of greater importance than speed.

If you are interested in the actual implementation of RK4 in the nMod n-Body model, then you may want to have the function move_particles_RK4 open in another tab while you work through this section. It is not required however, so if you're unfamiliar with C code you needn't bother.

Once again I have broken down the process into a series of tasks that must be completed in order to complete the integration. These once more map well to the implementation in nMod.

Task One: Initialisation

We start at the beginning of the step by obtaining all the gravitational interaction between the particles. This represents the starting velocities for RK4 integration.

We obtain the gravitational effects, convert them to velocity changes, and store the revised particle velocities found at this point as V1

Task Two: Advancing to the First Mid Point

Using the velocities obtained in task one, we advance to a point mid way between the total step size. Thus for a total step size of 30 seconds, we would advance for 15 seconds, using the velocities calculated in task one.

Once at this mid point, We again obtain the gravitational effects, convert them to velocity changes, and store the revised particle velocities found at this point as V2

Task Three: Returning to the Starting Position

We now discard the new positional data from task two, and return to the starting position. This being the positions occupied by all particles in the model during task one.

Task Three: Advancing to the second Mid Point

We initialise the particles in the model with the velocities found for them in task 2, and stored in V2, then advance them once again until half the time of the step has elapsed.

Once at this new mid point, we again obtain the revised gravitational effects, find the revised velocities for all particles in the model, and store these in V3

Task Four: Returning to the Starting Position

Again we discard the current positional information for all particles, and return them to the positions they occupied in task one.

Task Five: Advancing across the whole interval

We now initialise the particles in the model with the velocities stored in V3, and advance them across the entire step.

Once there, we again obtain the revised gravitational effects, find the revised velocities for all particles in the model, and store these in V4

Task Six: Returning to the Starting Position

For the final time, we discard the current positional information for all particles, and return them to the positions they occupied in task one.

Task Seven: Finding the Final Position

Here we combine the velocities stored in V1, V2, V3 and V4 in order to discover the smooth curve that will allow us to find the final positions of all particles in the model.

Taking the original positions of all particle in the model as YN, so the revised positions at the end of the step are YN+1, we have

YN+1 = YN + V1/6 + V2/3 + V3/3 + V4/6

Which gives us the desired new position and velocity data at the end of the step.

Summary

As you have seen, the RK4 method is a lot more involved than Midpoint integration. This additional complexity increases accuracy, but the greater amount of work means it is slower in comparison.

This concludes the Integration section. You might want to return to the User Manual.

Links

The Integration Methods in nMod: Part 2

The Integration Methods in nMod: Part 1

User Manual

Dr Carey Pridgeon 2008 - carey.pridgeon@gmail.com