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Krylov deferred correction accelerated method of lines transpose for parabolic problems. (English) Zbl 1134.65064
Summary: A new class of numerical methods for the accurate and efficient solution of parabolic partial differential equations is presented. Unlike the traditional method of lines (MoL), the new Krylov deferred correction (KDC) accelerated method of lines transpose (MoL\(^{\text T})\) 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\(^{\text T}\) technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time-stepsizes in long-time simulations.

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35K57 Reaction-diffusion equations
35Q55 NLS equations (nonlinear Schrödinger equations)
65H10 Numerical computation of solutions to systems of equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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