Could you please advise some programs or libraries for solving parabolic PDEs (or its systems) in 1D, 2D and 3D, for example, with the method of lines? The system of parabolic PDEs can be nonlinear in the right-hand side, for example, $\partial u/\partial t - a\Delta u + u^4 = 0$.
Or it is recommended to use only ODE solvers and construct the ODE system manually? What sparse ODE solvers can I use?
P.S.: Are there some methods of solving parabolic PDEs besides the method of lines? For example, one can discretize first by time and obtain elliptic equations at each time step. Is this method really slower than the method of lines? Does there exist something else?