Usually the Runge Kutta method uses 4 stages. What if I want to use an arbitrary number of stages like 300 stages, which is possible in some domains (e.g. with physics informed neural networks). Is there a general formula for the necessary constants?
I.e., I'm seeking sets of $c_i$, $b_i$, and $a_{ij}$ as described in the implicit RK section at https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods .
Gauß methods with $s$ stages have order 2s, Radau IIa methods have order 2s-1.
The order declines as one adds additional conditions. For instance that $A$ be lower triangular with the same number in every diagonal entry ((special) diagonally implicit RK methods).
For a more detailed overview see the slides PDFs in https://www.math.auckland.ac.nz/~butcher/ODE-book-2008/Tutorials/
While one can take 75 steps of RK4 and call it a 300 stage method, it will still have order 4. Using Richardson extrapolation like Burlisch-Stoer will increase the order, but is far from opimal. The order conditions for RK methods are polynomials of increasing degrees, making a general solution complicated to hopeless. With structured additional conditions methods in the order range 12-20 have been constructed. There do not exist real viable use cases for such high-order methods, cf. https://scicomp.stackexchange.com/questions/25581/why-are-higher-order-runge-kutta-methods-not-used-more-often