in this talk, we discuss a new class of numerical methods for the accurate and efficient integration of time dependent partial differential equations. unlike traditional method of lines $(mol)$, the new krylov deferred correction $(kdc)$ accelerated method of lines transpose $(mol^t)$ first discretizes the temporal direction using gaussian type nodes and spectral integration, and the resulting coupled elliptic system is solved iteratively using newton-krylov techniques such as newton-gmres method, in which each function evaluation is simply one low order time stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers. preliminary numerical experiments show that the kdc accelerated $mol^t$ technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time step sizes in long-time simulations. numerical experiments for parabolic type equations including the schrodinger equation will be discussed.