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

作     者：Hendrik Ranocha Katharina Ostaszewski Philip Heinisch Hendrik Ranocha;Katharina Ostaszewski;Philip Heinisch

作者机构：Institute Computational MathematicsTU BraunschweigUniversitatsplatz 238106 BrunswickGermany Extreme Computing Research Center(ECRC)Computer Electrical and Mathematical Science and Engineering Division(CEMSE)King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia Institut für Angewandte Numerische Wissenschaft e.V.(IANW)Bienroder Straae 338110 BrunswickGermany Institut für Geophysik und Extraterrestrische PhysikTU BraunschweigMendelssohnstraBe 338106 BrunswickGermany 

出 版 物：《Communications on Applied Mathematics and Computation》 (应用数学与计算数学学报（英文）)

年 卷 期：2020年第2卷第4期

页      面：581-611页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Summation by parts Vector calculus Helmholtz Hodge decomposition Mimetic properties Wave mode analysis 

摘      要：In this article,discrete variants of several results from vector calculus are studied for clas-sical finite difference summation by parts operators in two and three space *** is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations,which are characterised *** results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge ***,iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are pro-posed and applied *** numerical experiments,the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first-order partial differential *** 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.