RungeKuttaMethod.md

Gravitational acceleration

$x$ is a position vector here.

$$ f(x) = -\frac{x}{|x|^3} $$

Newton equation

$$ \frac{d^2 x}{dt^2} = f(x) = -\frac{x}{|x|^3} $$

Define an intermediate variable to make it first-order differetial equation $ v \equiv \frac{dx}{dt} $

$$ \frac{dv}{dt} = -\frac{x}{|x|^3} \\ \frac{dx}{dt} = v $$

Make the differential deltas

$$ \Delta v = \frac{x}{|x|^3} \Delta t \\ \Delta x = v \Delta t $$

Euler method

Formal definition:

$$ x_{n+1} = x_n + hf'(t_n, x_n) $$

In our case:

$$ v_{n+1} = v_n + h \frac{x}{|x|^3} \\ x_{t+1} = x_n + h v(t) $$

Modified Euler method (Midpoint method or Second-order Runge Kutta method)

Formal definition:

$$ \begin{align*} k_1 &= h f'(t_n, x_n) \\ k_2 &= h f'(t_n + \frac 12 h, x_n + \frac 12 k_1) \\ x_{n+1} &= x_n + k_2 \end{align*} $$

In our case:

$$ \begin{align*} \Delta v_1 &= h f(x) \\ \Delta x_1 &= h v(t) \\ \Delta v_2 &= h f(x + \Delta x_1 / 2) \\ \Delta x_2 &= h (v + \Delta v_1 / 2) \\ v(t+h) &= v(t) + \Delta v_2 \\ x(t+h) &= x(t) + \Delta x_2 \end{align*} $$