SciCode / 49.3 / Write a function that implements the 4th order accurate classical Runge-Kutta…

Background
The classical 4th order accurate time stepper is a multi-step time stepping algorithm that computes 4 intermediate values k₁, k₂, k₃, k₄ and combines them with the initial state according to

u_(out)=u_(in)+Δt/6(k₁+2k₂+2k₃+k₄)

where Δt is the time step size. This algorithm is 5th order accurate locally and 4th order accurate globally.

The textbook implementation of this algorithm is:

$\begin{aligned} & k_1=f\left(u_{in}, t_0\right) \\ & k_2=f\left(u_{in}+k_1 \frac{h}{2}, t_0+\frac{h}{2}\right) \\ & k_3=f\left(u_{in}+k_2 \frac{h}{2}, t_0+\frac{h}{2}\right) \\ & k_4=f\left(u_{in}+k_3 h, t_0+h\right)\end{aligned}$

Combine `uout = uin + dt/6*(k1+2*k2+2*k3+k4)`

Background
The classical 4th order accurate time stepper is a multi-step time stepping algorithm that computes 4 intermediate values $k_1$, $k_2$, $k_3$, $k_4$ and combines them with the initial state according to

$$
u_{out} = u_{in} + \Delta t/6 (k_1 + 2 k_2 + 2 k_3 + k_4)
$$

where $\Delta t$ is the time step size. This algorithm is 5th order accurate locally and 4th order accurate globally.

The textbook implementation of this algorithm is:

$\begin{aligned} & k_1=f\left(u_{in}, t_0\right) \\ & k_2=f\left(u_{in}+k_1 \frac{h}{2}, t_0+\frac{h}{2}\right) \\ & k_3=f\left(u_{in}+k_2 \frac{h}{2}, t_0+\frac{h}{2}\right) \\ & k_4=f\left(u_{in}+k_3 h, t_0+h\right)\end{aligned}$

Combine `uout = uin + dt/6*(k1+2*k2+2*k3+k4)`

Write a function that implements the 4th order accurate classical Runge-Kutta time integrator to evolve a set of particle locations and velocties forward in time subject to Newton's gravity. The function will take as input a state vector `uin` of size `N*6` for `N` particles containing in order the each particle's `x`, `y`, `z` location and its `vx`, `vy`, `vz` velocities as well as masses mass for each particle. Use the function `Nbody_RHS` described above to compute the right hand side of the evolution equation.

Write a function that implements the 4th order accurate classical Runge-Kutta time integrator to evolve a set of particle locations and velocties forward in time subject to Newton's gravity. The function will take as input a state vector `uin` of size `N*6` for `N` particles containing in order the each particle's `x`, `y`, `z` location and its `vx`, `vy`, `vz` velocities as well as masses $mass$ for each particle. Use the function `Nbody_RHS` described above to compute the right hand side of the evolution equation.