The Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshes

Handle URI:
http://hdl.handle.net/10754/566180
Title:
The Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshes
Authors:
Li, Jichun; Sun, Shuyu ( 0000-0002-3078-864X )
Abstract:
For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the (Formula presented.) norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.
KAUST Department:
Computational Transport Phenomena Lab; Physical Sciences and Engineering (PSE) Division
Publisher:
Springer Science + Business Media
Journal:
Journal of Scientific Computing
Issue Date:
2-Dec-2014
DOI:
10.1007/s10915-014-9963-5
Type:
Article
ISSN:
08857474
Appears in Collections:
Articles; Physical Sciences and Engineering (PSE) Division; Computational Transport Phenomena Lab

Full metadata record

DC FieldValue Language
dc.contributor.authorLi, Jichunen
dc.contributor.authorSun, Shuyuen
dc.date.accessioned2015-08-12T09:31:23Zen
dc.date.available2015-08-12T09:31:23Zen
dc.date.issued2014-12-02en
dc.identifier.issn08857474en
dc.identifier.doi10.1007/s10915-014-9963-5en
dc.identifier.urihttp://hdl.handle.net/10754/566180en
dc.description.abstractFor decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the (Formula presented.) norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.en
dc.publisherSpringer Science + Business Mediaen
dc.subjectMAC schemeen
dc.subjectNon-uniform rectangular gridsen
dc.subjectStaggered gridsen
dc.subjectStokes equationsen
dc.titleThe Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshesen
dc.typeArticleen
dc.contributor.departmentComputational Transport Phenomena Laben
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.identifier.journalJournal of Scientific Computingen
dc.contributor.institutionDepartment of Mathematical Sciences, University of Nevada Las VegasLas Vegas, NV, United Statesen
kaust.authorSun, Shuyuen
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.