Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators
- PDF / 3,669,699 Bytes
- 31 Pages / 439.37 x 666.142 pts Page_size
- 81 Downloads / 156 Views
Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators Hendrik Ranocha1,2,4 · Katharina Ostaszewski3,4 · Philip Heinisch3,4 Received: 23 August 2019 / Revised: 1 December 2019 / Accepted: 3 December 2019 © Shanghai University 2020
Abstract In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields 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 first-order partial differential 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. Keywords Summation by parts · Vector calculus · Helmholtz Hodge decomposition · Mimetic properties · Wave mode analysis Mathematics Subject Classification 65N06 · 65M06 · 65N35 · 65M70 · 65Z05 * Hendrik Ranocha h.ranocha@tu‑bs.de; [email protected]
Katharina Ostaszewski k.ostaszewski@tu‑bs.de; [email protected]
Philip Heinisch p.heinisch@tu‑bs.de; [email protected]
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
13
Vol.:(0123456789)
Communications on Applied Mathematics and Computation
1 Introduction The Helmholtz Hodge decomposition of a vector field into irrotational and solenoidal components and their respective scalar and vector potentials is a classical result that appears in many different variants both in the traditional fields of mathematics and physics and more recently in applied sciences like medical imaging [53]. Especially in the context of classical electromagnetism and plasma physics, the Helmholtz Hodge decomposition has been used for many years to help analyse turbulent velocity fields [5, 28] or separate current systems into source-free and irrotational components [19–22]. Numerical implementations can be useful
Data Loading...
