Ch. 12.3 1st Order ODE (RK4 Method)

An object in free fall under the influence of gravity, with constant acceleration: \(a = 9.81 \, \text{m/s}^2\)

$$ \frac{dv}{dt}=a ;\quad v(t=0)=0 $$

The following code implements the Runge-Kutta 4th order (RK4) method to calculate the velocity of the object over a time period.

Runge-Kutta 4th Order (RK4) Method

The RK4 method is applied to a differential equation of the form:

$$ \frac{dy}{dt} = f(t, y) $$

with an initial condition \( y(t_0) = y_0 \) over a time interval \([t_0, t_{\text{end}}]\). The method proceeds as follows:

Steps of RK4

For each time step \( h \), the RK4 method calculates four slopes:

1. Compute \( k_1 \): This is the initial slope, calculated at the beginning of the interval.

   $$ k_1 = f(t, y) $$

2. Compute \( k_2 \): This slope is calculated at the midpoint, using \( k_1 \) to approximate the halfway position.

   $$ k_2 = f\left(t + \frac{h}{2}, y + \frac{h}{2} \cdot k_1\right) $$

3. Compute \( k_3 \): Similar to \( k_2 \), but uses the slope from \( k_2 \) to further refine the midpoint estimate.

   $$ k_3 = f\left(t + \frac{h}{2}, y + \frac{h}{2} \cdot k_2\right) $$

4. Compute \( k_4 \): This final slope is calculated at the end of the interval.

   $$ k_4 = f(t + h, y + h \cdot k_3) $$

Update Rule

The next value, \( y_{\text{next}} \), is computed as a weighted average of these slopes:

$$ y_{\text{next}} = y + \frac{h}{6} \cdot (k_1 + 2 \cdot k_2 + 2 \cdot k_3 + k_4) $$