Hexagonal grid methods with applications to partial differential equations

Lee, D.; Tien, Hui-Chun; Luo, Cunjin; Luk, Hsiang‐Ning

doi:10.1080/00207160.2013.864392

Cited by 14 publications

(16 citation statements)

References 15 publications

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“…The locally second-order quadrature actually yields a (global) second-order approximation, and a fourthorder compact scheme is obtained with more effort [2].…”

Section: Leementioning

confidence: 99%

“…In the study of the origin of U-wave in ECG (electrocardiogram) [1], hexagonal subregions are adopted, but only pure algebraic relations are considered. In numerical simulation [2] of human heart electrophysiology, Hex FV based finite difference (FD) methods are developed and analyzed, leading to discrete ordinary and compact seven-point schemes of the standard Laplacian with applications to Poisson equations and reaction-diffusion systems. The work [2] showed advantages of Hex FVs over square grids in studies of wave phenomena propagated in curved domains, while the theoretic and numerical investigations were extended in a separate work to semi-linear Poisson equations involving anisotropic Laplacian on nets of regular hexagons of two specific orientations.…”

Section: Introductionmentioning

confidence: 99%

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Complete solution to seven-point schemes of discrete anisotropic Laplacian on regular hexagons

Lee

2015

APPL ANAL DISCRETE M

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“…The locally second-order quadrature actually yields a (global) second-order approximation, and a fourthorder compact scheme is obtained with more effort [2].…”

Section: Leementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Complete solution to seven-point schemes of discrete anisotropic Laplacian on regular hexagons

Lee

2015

APPL ANAL DISCRETE M

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“…The analysis of solving Laplacian related applications on net of (regular) hexagons may be based on series method (Lee, Tien, Luo & Luk, 2014), M-matrix theory (Lee, Tien, Luo & Luk, 2014) and (Lee, August 2017), functional interpolation (Lee, 2015), or Fourier vectors (Lee, August 2017). The current work discusses the Laplacian through discrete symmetric boundary condition (Sections 3 and 4).…”

Section: Introductionmentioning

confidence: 99%

“…We note for applications that a two-dimensional irregular domain may be approximated by a sequence of (not necessarily Cartesian) nets of hexagons. Actually, our work in numerical modeling of ECG depends on this (Algorithm 1 in (Lee,Tien,Luo & Luk, 2014)). (i, j) (1.5i − 0.5)r 2ih (2i − 1)h cy (i, j) 2 jh (2 j − 1)h (1.5 j − 0.5)r Table 2.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Boundary Condition for Laplacian on Net of Regular Hexagons

Lee¹

2017

JMR

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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.

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“…However, this is a purely algebraic approach to obtaining ECG phenomena. With the inclusion of a reaction-diffusion system, numerical simulation of cardiac electrophysiology is conducted [8] in a reversed C-type domain which is approximated by hexagonal finite volumes. We consider in this work numerical cubature on regular hexagons and develop a fourth-order seven-point rule.…”

Section: Introductionmentioning

confidence: 99%