Motivation:
I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form
$$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$
where $'=\frac{d}{dr}$ and $\dot{} =\frac{d}{dt}$. (The expression for the function $f$ is long to I don't include it, but it contains terms like $\dot{u}^nu'^m$.)
I need to solve this numerically, and I have done this by defining the 'velocity' $Q = \frac{du}{dt}$ so that we can write the system as
$$\dot{Q} = f(t, Q, Q', u' ,u'')$$ $$\dot{u} = Q$$
which I then discretize on a grid (for the space) and evolve in time using a a staggered leapfrog:
$$Q_{n+1/2} = Q_{n-1/2} + f(t_n, Q_n, Q'_n, u'_n ,u''_n)\Delta t$$ $$u_{n+1} = u_n + Q_{n+1/2}\Delta t$$
This works fine for many applications, but I have some problems with getting this to work in all situations I'm interested in.
Question
I'm therefore looking for alternative methods to solve this equation. I welcome any suggestions. I have been thinking about Newton-Gauss-Seidel, but it's pretty computationally expensive so I'm hoping there are other methods I could try first. Also, if there is someone that has experience with numerically solving such equations it would be great to get some input on methods that are known to work well in practice.
