[SciPy-Dev] Proposal for a new ODE Solver

[Laurin Steidle]

Hi all,

I opened an github issue [1] asking the following and was recommended to 
post it here.

I am *proposing to add a new ODE solver* to the list of solvers 
available in solve_ivp.
The solver I propose is the so called "*modified Patanka-Runge-Kutta" 
(MPRK)* [2] scheme.
MPRK is a integration scheme that satisfies the condition of 
'unconditional positivity',
a requirement for many ODEs especially in the field of biochemical models.
A simple and famous example of such a ODE being the Lotka-Volterra 
equations.

So far there exists no such solver in scipy.
Hence, a ODE solution with the present solvers can become "unphysical" 
and therefore invalid if the solution at a any time-step becomes 
negative due to numerical errors.

If this is something the community agrees on adding I am happy to write 
the implementation.
However, I would like to ask for assistance by a more experienced scipy 
dev as I am new to scipy.

Cheers,
Laurin

[1] https://github.com/scipy/scipy/issues/12766
[2] https://doi.org/10.1016/S0168-9274(03)00101-6

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