Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Diference Summation by Parts Operators

doi:10.1007/s42967-019-00057-2

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (4): 581-611.doi: 10.1007/s42967-019-00057-2

• ORIGINAL PAPER • 上一篇

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Diference Summation by Parts Operators

Hendrik Ranocha^1,2,4, Katharina Ostaszewski^3,4, Philip Heinisch^3,4

1 Institute Computational Mathematics, TU Braunschweig, Universitätsplatz 2, 38106 Brunswick, Germany;
2 Present Address:Extreme Computing Research Center(ECRC), Computer Electrical and Mathematical Science and Engineering Division(CEMSE), King Abdullah University of Science and Technology(KAUST), Thuwal 23955-6900, Saudi Arabia;
3 Institut für Geophysik und Extraterrestrische Physik, TU Braunschweig, Mendelssohnstraße 3, 38106 Brunswick, Germany;
4 Institut für Angewandte Numerische Wissenschaft e. V. (IANW), Bienroder Straße 3, 38110 Brunswick, Germany

收稿日期:2019-08-23 修回日期:2019-12-01 发布日期:2020-09-11
通讯作者: Hendrik Ranocha, Katharina Ostaszewski, Philip Heinisch E-mail:h.ranocha@tu-bs.de,h.ranocha@ianw.de;k.ostaszewski@tu-bs.de,k.ostaszewski@ianw.de;p.heinisch@tu-bs.de,p.heinisch@ianw.de

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Diference Summation by Parts Operators

Hendrik Ranocha^1,2,4, Katharina Ostaszewski^3,4, Philip Heinisch^3,4

1 Institute Computational Mathematics, TU Braunschweig, Universitätsplatz 2, 38106 Brunswick, Germany;
2 Present Address:Extreme Computing Research Center(ECRC), Computer Electrical and Mathematical Science and Engineering Division(CEMSE), King Abdullah University of Science and Technology(KAUST), Thuwal 23955-6900, Saudi Arabia;
3 Institut für Geophysik und Extraterrestrische Physik, TU Braunschweig, Mendelssohnstraße 3, 38106 Brunswick, Germany;
4 Institut für Angewandte Numerische Wissenschaft e. V. (IANW), Bienroder Straße 3, 38110 Brunswick, Germany

Received:2019-08-23 Revised:2019-12-01 Published:2020-09-11
Contact: Hendrik Ranocha, Katharina Ostaszewski, Philip Heinisch E-mail:h.ranocha@tu-bs.de,h.ranocha@ianw.de;k.ostaszewski@tu-bs.de,k.ostaszewski@ianw.de;p.heinisch@tu-bs.de,p.heinisch@ianw.de

摘要/Abstract

摘要： In this article, discrete variants of several results from vector calculus are studied for classical fnite diference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector felds cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully. In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other frst-order partial diferential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

关键词: Summation by parts, Vector calculus, Helmholtz Hodge decomposition, Mimetic properties, Wave mode analysis

Abstract: In this article, discrete variants of several results from vector calculus are studied for classical fnite diference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector felds cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully. In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other frst-order partial diferential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

Key words: Summation by parts, Vector calculus, Helmholtz Hodge decomposition, Mimetic properties, Wave mode analysis

中图分类号:

Hendrik Ranocha, Katharina Ostaszewski, Philip Heinisch. Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Diference Summation by Parts Operators[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 581-611.

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Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Diference Summation by Parts Operators

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Diference Summation by Parts Operators

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