Compare solvers

[LevanBokeria]

This issue will track the efforts to compare different solvers of ODEs with regards to their accuracy and speed. We will try different packages and implicit vs explicit solvers.

[LevanBokeria]

@MaxBalmus could you let me and @crangelsmith know what sorts of different solvers you have tried already, and the reasoning behind choosing the solve_ivp function from scipy?

Harry was suggesting today we could compare implicit and explicit solvers for both speed and accuracy. And have a gold standard solution provided by CVOD perhaps, to which all else can be compared to.

[MaxBalmus]

I chose solve_ivp mostly because I was familiar with scipy, but yes I know there are few alternatives out there from numpy, sundials, jax, etc. solve_ivp itself is a collection of methods including:

• RK45 (stands for Runge-Kutta method of order 5(4) default, explicit)
• RK23 (explicit)
• DOP854 (explicit Runge-Kutta of order 8)
• Radau (implicit)
• BDF (implicit)
• LSODA (implicit)

I have marked the solver I at least tried at some point.

Essentially the main problem with solver is that we want to balance cost and stability. Our problem is "stiff" which means some solvers like RK23, RK45 and DOP853 might not be ideal. Radar and BDF are designed for stiff problems but are more expensive than the others. LSODA is adaptive, can alternate between the two types of solvers, such that it can be seen as a universal solver. Additionally it has an adaptive step size, that is it can choose to increase the temporal resolution where it perceive that this would be beneficial. I chose it because it appeared to reduce the proportion of unstable simulations.