1. An adaptive Runge-Kutta integrator   [Solution]#

In the exercise on integrating planetary orbits, we looked at a circular orbit. The same equations apply for eccentric orbits as well, if you choose different initial conditions. For example, you can set an initial distance $r=1+e$ where $e$ is the eccentricity, and a perpendicular velocity $v=\sqrt{(2/r)-1}$ (from energy conservation). Because an object on an eccentric orbit spends different amounts of time in different parts of the orbit, integrating with a constant time-step is not very efficient, particularly for large eccentricities.

To address this issue, implement an adaptive-step-size RK4 integrator and use it to integrate an eccentric orbit over a time $2\pi$ (so the orbit closes), keeping the relative error just below 1 part in ${10}^6$.

To choose the step size you can do the following:

1. Take an RK4 step with the current step size $h$ and compare that with the result you get if you instead take two steps with half the step-size $h/2$. The difference between the results gives you a measure of the error.

2. If the error is smaller than the desired tolerance, then you keep the result and increase $h$ by a factor of 2. If the error is larger than the desired tolerance, reject this step and try again with $h$ smaller by a factor of 2.

3. Repeat. You will need to adjust the final step-size so that your final value of time is exactly at $t=2\pi$.

Plot the orbit to make sure you are indeed getting an ellipse and that the orbit goes back to its starting point after a time $2\pi$. Compare the number of steps you need to take to get an accuracy of ${10}^{-6}$ for a full $e=0.9$ orbit with adaptive step size and with constant step size.

2. Method of lines   [Solution]#

The method of lines is a way to do time-evolution of partial differential equations. For example, consider the thermal diffusion equation

where we set the thermal diffusivity to 1 for simplicity. We want to solve the following 1D diffusion problem: A piece of metal is initially at a temperature $T=1$ everywhere. At time $t=0$, the end of the piece of metal at $x=1$ is set to a temperature $T=0$ and held at that value. The temperature at the other end $x=0$ is held constant at $T=1$. Calculate the temperature profile as a function of time $T(x,t)$.

In the method of lines, we transform the PDE into an ODE by finite differencing the spatial derivative on a grid in $x$:

where $T_i$ is the temperature at $x=x_i$. If there are $N$ grid points, we now have $N$ coupled-ODEs that we can integrate in time.

Use this method to solve the problem described above. You should write your own implicit integrator using the techniques discussed in the notes (do not use solve_ivp for this problem).

You will need to take care at the boundaries: the values of $T_1$ and $T_N$ need to be held fixed according to the boundary conditions, so they obey $d{T}_1/dt=d{T}_N/dt=0$ rather than equation (18).

Plot the temperature profile $T(x)$ at different times and discuss whether it shows the behavior you expect.