Yet-another RK4 n-body problem

[pretzsp]

Homework Statement

Implementing the 4th order Runge-Kutta method for an n-body problem

Homework Equations

The RK4 method for the IVP y' = f(t,y), y(t0) = y0, is given by
y_(n+1) = y_n + (1/6)(k1 + 2 k2 +2 k3 + k4)
t_(n+1) = t_n + h

where

k1 = h f(t_n,y_n)
k2 = h f(t_n+ h/2, y_n + k1/2)
k3 = h f(t_n+ h/2, y_n + k2/2)
k4 = h f(t_n+ h/2, y_n + k3)

The Attempt at a Solution

I'm trying to simulate some n-body problems using the Runge-Kutta 4th Order method on Mathematica. I have something that works, for just the most simple problem of one body orbiting about a fixed mass.

step[data_] := (
t = data[[1]]; kx1 = data[[2]]; kx2 = data[[3]]; kx3 = data[[4]]; kx4 = data[[5]]; kv1 = data[[6]]; kv2 = data[[7]]; kv3 = data[[8]]; kv4 = data[[9]]; v = data[[10]]; x = data[[11]];

{t + h,

h ((1/m2) GravForce[m1, {0, 0}, m2, x]),
h ((1/m2) GravForce[m1, {0, 0}, m2,
x + k1/2]),
ht ((1/m2) GravForce[m1, {0, 0}, m2,
x + k2/2]),
h ((1/m2) GravForce[m1, {0, 0}, m2,
x + k3]),

h v,
h (v + k1 /2),
h (v + k2/2),
h (v + k3),

v + (1/6) (k1 + 2 k2 + 2 k3 + k4),
x + (1/6) (vk1 + 2 vk2 + 2 vk3 + vk4)

}

But I'm fairly sure that this is entirely wrong despite the fact that it gives me a "correct" result - for one thing, it's not significantly more accurate with the same value of h than Euler's Method. The other is that the functions themselves look completely and utterly wrong.

Could someone help me out here? This is really spinning my head around.

I'm not really looking for help with Mathematica, just more of a general guide on how Runge-Kutta is supposed to work for a system like this.

 

[anathema]

Hi there,

I can understand your confusion with implementing the RK4 method for an n-body problem. It can be quite challenging to code and understand, but I'll try to break it down for you.

First, let's look at the general steps for the RK4 method:

1. Define an initial value problem (IVP) with an initial condition y(t0) = y0.
2. Choose a step size h.
3. Use the RK4 formula to calculate the next values of y and t.
4. Repeat until you reach the desired final value of t.

Now, for an n-body problem, we have multiple variables (position and velocity) for each body, so our IVP will have multiple equations. Let's say we have n bodies, then our IVP will look something like this:

y' = f(t,y), where y = [x1, y1, vx1, vy1, x2, y2, vx2, vy2, ..., xn, yn, vxn, vyn]

Here, x and y represent the position coordinates and vx and vy represent the velocity components for each body. Now, we can use the RK4 method to solve this IVP and get the updated values for all the variables at each step.

Let's say we have n bodies, then our RK4 formula will look like this:

y_(n+1) = y_n + (1/6)(k1 + 2 k2 +2 k3 + k4)

where y_n is the current value of the variables and k1, k2, k3, and k4 are calculated using the following equations:

k1 = h f(t_n,y_n)
k2 = h f(t_n+ h/2, y_n + k1/2)
k3 = h f(t_n+ h/2, y_n + k2/2)
k4 = h f(t_n+ h/2, y_n + k3)

Here, h is the step size and f(t,y) represents the equations of motion for each body. So, for each step, we need to calculate the values of k1, k2, k3, and k4 for all the variables and use them to update the values of y at the next step.

I hope this helps in understanding the general concept of implementing the RK4 method for an n-body problem. You can use this as a guide for coding