This article is great, especially having some code to mull and reuse! Thank you.

(Sorry for this long post)

I’m doing a 3-space n-body gravitational sim and am starting to come to the unpleasant conclusion that with RK4 I’m going to have to compute the effects due to all n bodies on each other all at once inside evaluate() and acceleration(), because I can’t isolate the effects of each as I could with Euler or a modifed Euler. If I don’t update all the bodies in each evaluate() call, I’ll see the 2nd, 3rd, and 4th evaluate() calls start to accumulate excess error because the other body positions are stale. Your comment above for the damped spring:

> you integrate both bodies together, ideally you have a single method that returns an array of forces to apply to the points — you cant update each body independently with RK4

seems to support this gloomy conclusion. In a high-gradient field, i.e. when near another massive body, this will become quite important I suspect. I’ll have to fully compute updated state for each of the other n-1 bodies including inside each of the 4 evaluate() calls in integrate(), much as you describe above for the spring case. So I end up doing them all at once.

evaluate() and accelerate() look like becoming much more complex, and computationally quite expensive for large n. integrate() will have some loops, but will not be especially troublesome, I believe.

I think the changes needed, in rough outline, are:

- turn State and Derivative into arrays[n] of structs, each struct containing 6 components (x, y, z, vx, vy, vz), with n being the number of bodies, and have caller pass states[] into integrate() along with t and dt, also making available masses[] and G somehow, maybe as parameters. More on integrate() later.

- accelerate() needs to compute G*mj/r^2 (to get acceleration mi on body bi) for each pair of bodies with masses and return the results in outputs[n]

- evaluate() needs to update all the elements of states[] with the initial.x + d.dx*dt and initial.vx + d.dv*dt computations for each of the 3 axes, initialize outputs[].dx, .dy, and.dz, then invoke acceleration() on states[], which will copy results to outputs[] for evaluate().

- integrate() needs to make the 4 calls to evaluate(), starting with a Derivatives[] initialized to {0.0, 0.0, 0.0, 0.0, 0.0, 0.0} in each array element. Then it can serially (at last, no parallel computations!) compute the dxdt, dydt, dzdt, dvxdt, dvydt, and dvzdt values for each array element and update that element’s state. values.

Does this sound about right?

Greetings from Germany