SciCode / 78.2 / Write a fourth-order fixed-time-step Runge-Kutta (RK4) integrator from scratch…

Background

Given an initial value problem of the form:

(dy)/(dt)=f(t,y), y(t₀)=y₀

the RK4 method updates the solution y from time t to t+Δt using the following steps:

**2.1 Calculate intermediate slopes:**
   - k₁=f(t,y)
   - k₂=f(t+(Δt)/(2),y+(k₁Δt)/(2))
   - k₃=f(t+(Δt)/(2),y+(k₂Δt)/(2))
   - k₄=f(t+Δt,y+k₃Δt)

**2.2 Update the solution:**

   y(t+Δt)=y(t)+(Δt)/(6)(k₁+2k₂+2k₃+k₄)

In our case, *y* represents the state vector of the system, which includes the angle *θ* and the angular velocity *ω*.

Background

Given an initial value problem of the form:

$$
\frac{dy}{dt} = f(t, y), \quad y(t_0) = y_0
$$

the RK4 method updates the solution $y$ from time $t$ to $t + \Delta t$ using the following steps:

**2.1 Calculate intermediate slopes:**
   - $k_1 = f(t, y)$
   - $k_2 = f(t + \frac{\Delta t}{2}, y + \frac{k_1 \Delta t}{2})$
   - $k_3 = f(t + \frac{\Delta t}{2}, y + \frac{k_2 \Delta t}{2})$
   - $k_4 = f(t + \Delta t, y + k_3 \Delta t)$

**2.2 Update the solution:**

   $$
   y(t + \Delta t) = y(t) + \frac{\Delta t}{6} (k_1 + 2k_2 + 2k_3 + k_4)
   $$

In our case, *$y$* represents the state vector of the system, which includes the angle *$\theta$* and the angular velocity *$\omega$*.

Write a fourth-order fixed-time-step Runge-Kutta (RK4) integrator from scratch in Python, to calculate θ and ω at any certain timepoint. The output of this procedure — a series of state vectors representing the n-point state-space trajectory emanating from ~x(t0). Do not use any canned numerical integration routines, commands, functions.

Write a fourth-order fixed-time-step Runge-Kutta (RK4) integrator from scratch in Python, to calculate $\theta$ and $\omega$ at any certain timepoint. The output of this procedure — a series of state vectors representing the n-point state-space trajectory emanating from ~x(t0). Do not use any canned numerical integration routines, commands, functions.