Abstract
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
{\bf \it Krylov deferred correction (KDC) accelerated method of lines
transpose (MoL^T)} 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 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.