# Numerically solving systems of about 100 ODEs

I am looking to solve large systems of non-linear ODEs. There appears to be a very large list of methods available varying in complexity, and I have a hard time searching through them and picking one. Are any of these methods preferred for large systems? Both speed and accuracy play large roles, so I'm hoping that there are methods that are in general considered to be better for large systems.

Some additional information: The systems usually consist of ~100 ODEs that are quite heavily linked, usually consisting of a lot of quartic terms. (2-loop renormalization group equations)

Thanks

• An invariant of an equation is something that doesn't change as time passes if there are no external forces (and that changes in predictable ways if there are external forces). An example is the energy, or the angular momentum, or the linear momentum. Even if you consider rather complex equations, such as the many-body gravitational interaction between point masses where $\ddot x_i(t) = - G \sum_{j\neq i} \frac{m_im_j}{|x_i(t)-x_j(t)|^2}$, all of the above are conserved -- in other words, they are invariants. Most ODE integrators do not preserve invariants, but some do. – Wolfgang Bangerth Sep 21 '13 at 0:55