Tensor product generalized ADI methods for separable elliptic problems
SIAM Journal on Numerical Analysis
Asymptotically stable fourth-order accurate schemes for the diffusion equation on complex shapes
Journal of Computational Physics
SIAM Journal on Numerical Analysis
On a High Order Numerical Method for Solving Partial Differential Equations in Complex Geometries
Journal of Scientific Computing
An integral evolution formula for the wave equation
Journal of Computational Physics
High Order Schemes for Resolving Waves: Number of Points per Wavelength
Journal of Scientific Computing
A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds
Journal of Computational Physics
A Modified Fourier–Galerkin Method for the Poisson and Helmholtz Equations
Journal of Scientific Computing
High order marching schemes for the wave equation in complex geometry
Journal of Computational Physics
Stable, high-order discretization for evolution of the wave equation in 2 + 1 dimensions
Journal of Computational Physics
Finite Differences And Partial Differential Equations
Finite Differences And Partial Differential Equations
Bounded Error Schemes for the Wave Equation on Complex Domains
Journal of Scientific Computing
Numerical Solution of Partial Differential Equations: An Introduction
Numerical Solution of Partial Differential Equations: An Introduction
Stability and accuracy of time-extrapolated ADI-FDTD methods for solving wave equations
Journal of Computational and Applied Mathematics
Journal of Computational Physics
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Journal of Computational Physics
Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws
Journal of Computational Physics
Journal of Computational Physics
A Fast Algorithm for Fourier Continuation
SIAM Journal on Scientific Computing
Approximation error in regularized SVD-based Fourier continuations
Applied Numerical Mathematics
On the resolution power of Fourier extensions for oscillatory functions
Journal of Computational and Applied Mathematics
Spatially Dispersionless, Unconditionally Stable FC---AD Solvers for Variable-Coefficient PDEs
Journal of Scientific Computing
Hi-index | 31.47 |
A new PDE solver was introduced recently, in Part I of this two-paper sequence, on the basis of two main concepts: the well-known Alternating Direction Implicit (ADI) approach, on one hand, and a certain ''Fourier Continuation'' (FC) method for the resolution of the Gibbs phenomenon, on the other. Unlike previous alternating direction methods of order higher than one, which only deliver unconditional stability for rectangular domains, the new high-order FC-AD (Fourier-Continuation Alternating-Direction) algorithm yields unconditional stability for general domains-at an O(Nlog(N)) cost per time-step for an N point spatial discretization grid. In the present contribution we provide an overall theoretical discussion of the FC-AD approach and we extend the FC-AD methodology to linear hyperbolic PDEs. In particular, we study the convergence properties of the newly introduced FC(Gram) Fourier Continuation method for both approximation of general functions and solution of the alternating-direction ODEs. We also present (for parabolic PDEs on general domains, and, thus, for our associated elliptic solvers) a stability criterion which, when satisfied, ensures unconditional stability of the FC-AD algorithm. Use of this criterion in conjunction with numerical evaluation of a series of singular values (of the alternating-direction discrete one-dimensional operators) suggests clearly that the fifth-order accurate class of parabolic and elliptic FC-AD solvers we propose is indeed unconditionally stable for all smooth spatial domains and for arbitrarily fine discretizations. To illustrate the FC-AD methodology in the hyperbolic PDE context, finally, we present an example concerning the Wave Equation-demonstrating sixth-order spatial and fourth-order temporal accuracy, as well as a complete absence of the debilitating ''dispersion error'', also known as ''pollution error'', that arises as finite-difference and finite-element solvers are applied to solution of wave propagation problems.