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High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

Bruno, Oscar P. and Lyon, Mark (2010) High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements. Journal of Computational Physics, 229 (6). pp. 2009-2033. ISSN 0021-9991.

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We introduce a new methodology for the numerical solution of Partial Differential Equations in general spatial domains: our algorithms are based on the use of the well-known Alternating Direction Implicit (ADI) approach in conjunction with a certain "Fourier continuation" (FC) method for the resolution of the Gibbs phenomenon. Unlike previous alternating direction methods of order higher than one, which can only deliver unconditional stability for rectangular domains, the present high-order algorithms possess the desirable property of unconditional stability for general domains; the computational time required by our algorithms to advance a solution by one time-step, in turn, grows in an essentially linear manner with the number of spatial discretization points used. In this paper we demonstrate the FC-AD methodology through a variety of examples concerning the Heat and Laplace Equations in two and three-dimensional domains with smooth boundaries. Applications of the FC-AD methodology to Hyperbolic PDEs together with a theoretical discussion of the method will be put forth in a subsequent contribution. The numerical examples presented in this text demonstrate the unconditional stability and high-order convergence of the proposed algorithms, as well the very significant improvements they can provide (in one of our examples we demonstrate a one thousand improvement factor) over the computing times required by some of the most efficient alternative general-domain solvers.

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Bruno, Oscar P.0000-0001-8369-3014
Additional Information:© 2009 Elsevier Inc. Received 17 April 2009. Received in revised form 25 October 2009. Accepted 11 November 2009. Available online 18 November 2009. We gratefully acknowledge support by the Air Force Office of Scientific Research and the National Science Foundation.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)UNSPECIFIED
Subject Keywords:Spectral method; Complex geometry; Unconditional stability; Fourier series; Fourier continuation; ADI; Partial Differential Equation; Numerical method
Issue or Number:6
Record Number:CaltechAUTHORS:20100324-103102353
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:17785
Deposited By: Joy Painter
Deposited On:29 Mar 2010 22:54
Last Modified:09 Mar 2020 13:18

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