Peano—A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids

Handle URI:
http://hdl.handle.net/10754/599153
Title:
Peano—A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids
Authors:
Weinzierl, Tobias; Mehl, Miriam
Abstract:
Almost all approaches to solving partial differential equations (PDEs) are based upon a spatial discretization of the computational domain-a grid. This paper presents an algorithm to generate, store, and traverse a hierarchy of d-dimensional Cartesian grids represented by a (k = 3)- spacetree, a generalization of the well-known octree concept, and it also shows the correctness of the approach. These grids may change their adaptive structure throughout the traversal. The algorithm uses 2d + 4 stacks as data structures for both cells and vertices, and the storage requirements for the pure grid reduce to one bit per vertex for both the complete grid connectivity structure and the multilevel grid relations. Since the traversal algorithm uses only stacks, the algorithm's cache hit rate is continually higher than 99.9 percent, and the runtime per vertex remains almost constant; i.e., it does not depend on the overall number of vertices or the adaptivity pattern. We use the algorithmic approach as the fundamental concept for a mesh management for d-dimensional PDEs and for a matrix-free PDE solver represented by a compact discrete 3 d-point operator. In the latter case, one can implement a Jacobi smoother, a Krylov solver, or a geometric multigrid scheme within the presented traversal scheme which inherits the low memory requirements and the good memory access characteristics directly. © 2011 Society for Industrial and Applied Mathematics.
Citation:
Weinzierl T, Mehl M (2011) Peano—A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids. SIAM Journal on Scientific Computing 33: 2732–2760. Available: http://dx.doi.org/10.1137/100799071.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Scientific Computing
KAUST Grant Number:
UK-c0020
Issue Date:
Jan-2011
DOI:
10.1137/100799071
Type:
Article
ISSN:
1064-8275; 1095-7197
Sponsors:
This work was supported by the InternationalGraduate School of Science and Engineering (IGSSE) and the Institute for Advanced Study (IAS)of the Technische Universitat Munchen. Furthermore, this publication is partially based on worksupported by Award No. UK-c0020, made by the King Abdullah University of Science and Technology(KAUST).
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorWeinzierl, Tobiasen
dc.contributor.authorMehl, Miriamen
dc.date.accessioned2016-02-25T13:53:52Zen
dc.date.available2016-02-25T13:53:52Zen
dc.date.issued2011-01en
dc.identifier.citationWeinzierl T, Mehl M (2011) Peano—A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids. SIAM Journal on Scientific Computing 33: 2732–2760. Available: http://dx.doi.org/10.1137/100799071.en
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/100799071en
dc.identifier.urihttp://hdl.handle.net/10754/599153en
dc.description.abstractAlmost all approaches to solving partial differential equations (PDEs) are based upon a spatial discretization of the computational domain-a grid. This paper presents an algorithm to generate, store, and traverse a hierarchy of d-dimensional Cartesian grids represented by a (k = 3)- spacetree, a generalization of the well-known octree concept, and it also shows the correctness of the approach. These grids may change their adaptive structure throughout the traversal. The algorithm uses 2d + 4 stacks as data structures for both cells and vertices, and the storage requirements for the pure grid reduce to one bit per vertex for both the complete grid connectivity structure and the multilevel grid relations. Since the traversal algorithm uses only stacks, the algorithm's cache hit rate is continually higher than 99.9 percent, and the runtime per vertex remains almost constant; i.e., it does not depend on the overall number of vertices or the adaptivity pattern. We use the algorithmic approach as the fundamental concept for a mesh management for d-dimensional PDEs and for a matrix-free PDE solver represented by a compact discrete 3 d-point operator. In the latter case, one can implement a Jacobi smoother, a Krylov solver, or a geometric multigrid scheme within the presented traversal scheme which inherits the low memory requirements and the good memory access characteristics directly. © 2011 Society for Industrial and Applied Mathematics.en
dc.description.sponsorshipThis work was supported by the InternationalGraduate School of Science and Engineering (IGSSE) and the Institute for Advanced Study (IAS)of the Technische Universitat Munchen. Furthermore, this publication is partially based on worksupported by Award No. UK-c0020, made by the King Abdullah University of Science and Technology(KAUST).en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subjectAdaptive Cartesian griden
dc.subjectCache efficiencyen
dc.subjectMultiscaleen
dc.subjectOctreeen
dc.subjectPartial differential equationen
dc.subjectSpace-filling curveen
dc.subjectSpacetreeen
dc.titlePeano—A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Gridsen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.contributor.institutionTechnische Universitat Munchen, Munich, Germanyen
kaust.grant.numberUK-c0020en
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