Second Order Accuracy of the 4-Point Hexagonal Net Grid
Finite Difference Scheme for Solving the 2D Helmholtz Equation

Eric S. Carlson and Haiwei Sun
Department of Chemical Engineering
University of Alabama
P. O. Box 870203
Tuscaloosa, AL 35487-0203, USA

and
Duane H. Smith
United States Department of Energy
National Energy Technology Laboratory
Morgantown, WV 26507, USA

and
Jun Zhang
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506-0046, USA

Abstract

Finite difference solutions of the Helmholtz equation that use the 4-point stencil of the hexagonal net grid exhibit global $O(h^2)$ accuracy, despite a local truncation error that is $O(h)$. We present a proof for why this happens, and provide the results of a number of computational tests to verify the second order behavior. Furthermore, the examples suggest that the 4-point solutions have comparable accuracy to those solutions using the 7-point stencil of the triangular net grid.


Key words: Helmholtz equation, hexagonal net grid, triangular net grid, hexagonal grid, triangular grid, second order scheme, finite difference

Mathematics Subject Classification: 65F10, 65N06, 65N22, 65N5 5, 76D07


Download the compressed postscript file hexgrid2.ps.gz, or the PDF file hexgrid2.pdf.
Technical Report No. 378-03, Department of Computer Science, University of Kentucky, Lexington, KY, 2003.