On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions

Handle URI:
http://hdl.handle.net/10754/599049
Title:
On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions
Authors:
Settle, Sean O.; Douglas, Craig C.; Kim, Imbunm; Sheen, Dongwoo
Abstract:
The primary aim of this paper is to answer the question, What are the highest-order five- or nine-point compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one- and two-dimensional Poisson equation on uniform, quasi-uniform, and nonuniform face-to-face hyperrectangular grids and directly prove the existence or nonexistence of their highest-order local accuracies. Our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights. For the one-dimensional problem, the derivation using the three-point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high-order local accuracy. However, for the two-dimensional problem, the derivation using the corresponding five-point stencil on uniform and quasi-uniform grids yields a scheme with at most second-order local accuracy, and on nonuniform grids yields at most first-order local accuracy. When expanding the five-point stencil to the nine-point stencil, the derivation using the nine-point stencil on uniform grids yields at most sixth-order local accuracy, but on quasi- and nonuniform grids yields at most fourth- and third-order local accuracy, respectively. © 2013 Society for Industrial and Applied Mathematics.
Citation:
Settle SO, Douglas CC, Kim I, Sheen D (2013) On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions. SIAM J Numer Anal 51: 2470–2490. Available: http://dx.doi.org/10.1137/120875570.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Numerical Analysis
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
Jan-2013
DOI:
10.1137/120875570
Type:
Article
ISSN:
0036-1429; 1095-7170
Sponsors:
The first, third, and fourth authors' research was partially supported by NRF of Korea (2011-0000344).This author's research was supported in part by NSF grants CNS-1018072 and EPS-1135483 and award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).
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Full metadata record

DC FieldValue Language
dc.contributor.authorSettle, Sean O.en
dc.contributor.authorDouglas, Craig C.en
dc.contributor.authorKim, Imbunmen
dc.contributor.authorSheen, Dongwooen
dc.date.accessioned2016-02-25T13:51:52Zen
dc.date.available2016-02-25T13:51:52Zen
dc.date.issued2013-01en
dc.identifier.citationSettle SO, Douglas CC, Kim I, Sheen D (2013) On the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditions. SIAM J Numer Anal 51: 2470–2490. Available: http://dx.doi.org/10.1137/120875570.en
dc.identifier.issn0036-1429en
dc.identifier.issn1095-7170en
dc.identifier.doi10.1137/120875570en
dc.identifier.urihttp://hdl.handle.net/10754/599049en
dc.description.abstractThe primary aim of this paper is to answer the question, What are the highest-order five- or nine-point compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one- and two-dimensional Poisson equation on uniform, quasi-uniform, and nonuniform face-to-face hyperrectangular grids and directly prove the existence or nonexistence of their highest-order local accuracies. Our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights. For the one-dimensional problem, the derivation using the three-point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high-order local accuracy. However, for the two-dimensional problem, the derivation using the corresponding five-point stencil on uniform and quasi-uniform grids yields a scheme with at most second-order local accuracy, and on nonuniform grids yields at most first-order local accuracy. When expanding the five-point stencil to the nine-point stencil, the derivation using the nine-point stencil on uniform grids yields at most sixth-order local accuracy, but on quasi- and nonuniform grids yields at most fourth- and third-order local accuracy, respectively. © 2013 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThe first, third, and fourth authors' research was partially supported by NRF of Korea (2011-0000344).This author's research was supported in part by NSF grants CNS-1018072 and EPS-1135483 and award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectCompact schemeen
dc.subjectFinite difference methoden
dc.subjectHermitian methoden
dc.subjectHighest-orderen
dc.subjectPoisson equationen
dc.titleOn the Derivation of Highest-Order Compact Finite Difference Schemes for the One- and Two-Dimensional Poisson Equation with Dirichlet Boundary Conditionsen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Numerical Analysisen
dc.contributor.institutionAltera Corporation, San Jose, United Statesen
dc.contributor.institutionUniversity of Wyoming, Laramie, United Statesen
dc.contributor.institutionSeoul National University, Seoul, South Koreaen
kaust.grant.numberKUS-C1-016-04en
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