At some day ago, I want to compute a specific example about Ricci flow. So, I try to compute the Ricci flow on $S^2\times S^2$, accord to the parameter of it. (I think the parameter is right, but seemingly not.)
In fact, I have computed out the Ricci tensor, connection, and balabla by software. At last, I get a system of nonlinear PDE, I really don't know how to get the solution. Besides, it has become very complex. So, I fail.
But I still want a non trivial example of Ricci flow, I think $T^3$ is a good start. But if I do it as before, I guess that, eventually, I will get a system of nonlinear PDE too.
So, I want to know, when studying Ricci flow on some manifold, what is the probable way? If I want to know the behaviour of $T^3$ under Ricci flow, what should I do?