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High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations

Lyon, Mark and Bruno, Oscar P. (2010) High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations. Journal of Computational Physics, 229 (9). pp. 3358-3381. ISSN 0021-9991. https://resolver.caltech.edu/CaltechAUTHORS:20100520-133258749

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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 O(N log(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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/j.jcp.2010.01.006 DOIArticle
ORCID:
AuthorORCID
Bruno, Oscar P.0000-0001-8369-3014
Additional Information:© 2010 Elsevier Inc. Received 17 April 2009; revised 1 January 2010; accepted 7 January 2010. Available online 20 January 2010. We gratefully acknowledge support by the Air Force Office of Scientific Research and the National Science Foundation.
Funders:
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)UNSPECIFIED
NSFUNSPECIFIED
Subject Keywords:Spectral method; Complex geometry; Unconditional stability; Fourier series; Fourier Continuation; ADI; Partial differential equation; Numerical method
Issue or Number:9
Record Number:CaltechAUTHORS:20100520-133258749
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20100520-133258749
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18367
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:21 May 2010 16:04
Last Modified:03 Oct 2019 01:41

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