GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Spectral integration and two-point boundary value problems
SIAM Journal on Numerical Analysis
A direct adaptive Poisson solver of arbitrary order accuracy
Journal of Computational Physics
A Fast Adaptive Numerical Method for Stiff Two-Point Boundary Value Problems
SIAM Journal on Scientific Computing
A New Fast-Multipole Accelerated Poisson Solver in Two Dimensions
SIAM Journal on Scientific Computing
On the A-Stability of Runge--Kutta Collocation Methods Based on Orthogonal Polynomials
SIAM Journal on Numerical Analysis
Jacobian-free Newton-Krylov methods: a survey of approaches and applications
Journal of Computational Physics
A New Parallel Kernel-Independent Fast Multipole Method
Proceedings of the 2003 ACM/IEEE conference on Supercomputing
A fast direct solver for boundary integral equations in two dimensions
Journal of Computational Physics
Accelerating the convergence of spectral deferred correction methods
Journal of Computational Physics
An adaptive fast solver for the modified Helmholtz equation in two dimensions
Journal of Computational Physics
A wideband fast multipole method for the Helmholtz equation in three dimensions
Journal of Computational Physics
Arbitrary order Krylov deferred correction methods for differential algebraic equations
Journal of Computational Physics
Is Gauss Quadrature Better than Clenshaw-Curtis?
SIAM Review
Journal of Computational Physics
Fast integral equation methods for Rothe's method applied to the isotropic heat equation
Computers & Mathematics with Applications
Stable and Spectrally Accurate Schemes for the Navier-Stokes Equations
SIAM Journal on Scientific Computing
An enhanced parareal algorithm based on the deferred correction methods for a stiff system
Journal of Computational and Applied Mathematics
Hi-index | 31.45 |
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(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-stepsizes in long-time simulations.