My final project in my Partial Differential Equations class involved studying one non-linear PDE in depth. In reading about my equation, I've realized that PDEs of 3 spatial variables can be re-written as an ODE if you assume spherical symmetry (which, for the vast majority of physical applications of this equation, you can).
I'm trying to motivate this change to an ODE, but the resulting ODE is non-linear and not analytically solvable (just like the original PDE). I want to claim that this is still useful because the methods for approximating ODEs (Runge-Kutta, mostly) are "better" than methods for approximating PDEs (finite element/difference/volume methods). However, I'm not familiar with these methods, and think the time learning them would be better spent on other parts of the project.
Is it a "generally accepted" claim that solving an ODE numerically is easier or more accurate than a PDE?
Numerical methods are often based around full derivatives and not partial derivatives. Of course "easier" is not a quantifiable term but there are more (accurate) methods of solving ODEs numerically than there are of solving PDEs. Furthermore more, what will validate your claim is a solid proof or if this is for a thesis than just "enough" examples.