If you fix the spatial discretization, then you are solving a fixed system of ODEs. The exact solution of that system is not the same as the exact solution of the PDE, of course, but you can test convergence to it just like you would with any ODE system.

If you fix the spatial discretization, then you are solving a fixed system of ODEs. The exact solution of that system is not the same as the exact solution of the PDE, of course, but you can test convergence to it just like you would with any ODE system.

Since the exact solution is usually too complicated to work out by hand, one typically uses a highly refined (in time) solution as a substitute.