High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations Mark Lyon a , Oscar P. Bruno b, * a University of New Hampshire, Department of Mathematics, Kingsbury Hall W348, NH 03824, United States b California Institute of Technology, Applied and Computational Mathematics, MC 217-50, 1200 East California Blvd., CA 91125, United States article info Article history: Received 17 April 2009 Received in revised form 1 January 2010 Accepted 7 January 2010 Available online 20 January 2010 Keywords: Spectral method Complex geometry Unconditional stability Fourier series Fourier Continuation ADI Partial differential equation Numerical method abstract 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 OðN logðNÞÞ cost per time-step for an N point spatial discretization grid. In the present contribution we provide an overall theoret- ical 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 intro- duced 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 gen- eral 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 accu- racy, 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. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction The FC-AD (Fourier-Continuation Alternating-Direction) methodology introduced in [1] (Part I of this two-paper se- quence) relies on two main elements: a novel spectral technique for general spatial domains (which is based on the one- dimensional Fourier Continuation method introduced in Part I) and the classical ADI approach pioneered by Douglas, Peac- eman and Rachford [2–6]. Unlike previous alternating direction methods of order higher than one, which only deliver uncon- 0021-9991/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2010.01.006 DOI of original article: 10.1016/j.jcp.2009.11.020 * Corresponding author. Tel.: +1 626 395 4548; fax: +1 626 578 0124. E-mail address: bruno@acm.caltech.edu (O.P. Bruno). Journal of Computational Physics 229 (2010) 3358–3381 Contents lists available at ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp