Abstract
Hexagonal grid methods are found useful in many research works, including numerical modeling in spherical coordinates, in atmospheric and ocean models, and simulation of electrical wave phenomena in cardiac tissues. Almost all of these used standard Laplacian and mostly on one configuration of regular hexagons. In this work, discrete symmetric boundary condition and energy product for anisotropic Laplacian are investigated firstly on general net of regular hexagons, and then generalized to its most extent in two- or three-dimensional cell-center finite difference applications up to the usage of symmetric stencil in central differences. For analysis of Laplacian related applications, this provides with an approach in addition to the M-matrix theory, series method, functional interpolations and Fourier vectors.
Highlights
Hexagonal (Hex) grid methods are of interest in many research studies: (Pickering,1986) on direct method, (Makarov, Mararov & Moskal’kov, 1993) giving a formula without proof, (Bystrytskyi & Mosklkov, 2001) on seven-point method on rectangular grid with explicit form of eigenpairs, (Zhou & Fulton, 2009) with periodic boundary condition (BC), (Heikes & Randall, 1995, part I,II) and (Heikes, Randall & Konor, 2013) on numerical modeling in spherical coordinates, on action potential in heart modeling via algebraic method without using diffusion in form of differential equation, (Nickovic,Gavrilov & Tosic, 2002) showing advantages of Hex grids over commonly used square grids for use in atmospheric and ocean models
In the article by (Lee,Tien,Luo and Luk,2014), Hex grid finite difference (FD) methods are derived in a finite volume (FV) approach involving standard Laplacian, and used in the simulation of electrical wave phenomena propagated in two-dimensional reversed-C type cardiac tissues, exhibiting both linear and spiral waves more efficiently than similar computation carried on rectangular FVs
As a practice for long time in the engineering literatures, the phrase symmetric boundary condition may mean, differently, that the computational domain is reduced by halving and the numerical BC on the virtual separating edge is of homogeneous Neumann type : (Kim & Huh, 2000), (Xu & Soares, 2013), and (Pal, Lan, Li, Hirleman & Ma, 2015)
Summary
Hexagonal (Hex) grid methods are of interest in many research studies: (Pickering,1986) on direct method, (Makarov, Mararov & Moskal’kov, 1993) giving a formula without proof, (Bystrytskyi & Mosklkov, 2001) on seven-point method on rectangular grid with explicit form of eigenpairs, (Zhou & Fulton, 2009) with periodic boundary condition (BC), (Heikes & Randall, 1995, part I,II) and (Heikes, Randall & Konor, 2013) on numerical modeling in spherical coordinates, (van Eck & Kors, 2005) on action potential in heart modeling via algebraic method without using diffusion in form of differential equation, (Nickovic,Gavrilov & Tosic, 2002) showing advantages of Hex grids over commonly used square grids for use in atmospheric and ocean models. As a practice for long time in the engineering literatures, the phrase symmetric boundary condition may mean, differently, that the computational domain is reduced by halving and the numerical BC on the virtual separating edge is of homogeneous Neumann type : (Kim & Huh, 2000), (Xu & Soares, 2013), and (Pal, Lan, Li, Hirleman & Ma, 2015).
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