Krylov deferred correction accelerated method of lines transpose for parabolic problems

In this paper, a new class of numerical methods for the accurate and efficient solutions of parabolic partial differential equations is presented. Unlike traditional method of lines (MoL), the new Krylov deferred correction (KDC) accelerated method of lines transpose  (MoLT)(MoLT) first discretizes the temporal direction using Gaussian type nodes and spectral integration, and symbolically applies low-order time marching schemes to form a preconditioned elliptic system, which is then solved iteratively using Newton–Krylov techniques such as Newton–GMRES or Newton–BiCGStab method. Each function evaluation in the Newton–Krylov method 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 MoLTMoLT technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time-stepsizes in long-time simulations.