Discretization Errors and Rotational Symmetry: The Laplacian Operator on Non-Hypercubical Lattices

Details Discretization Errors and Rotational Symmetry: The Laplacian Operator on Non-Hypercubical Lattices Discretization Errors and Rotational Symmetry: The Laplacian Operator on Non-Hypercubical Lattices

1998-10-23

arxiv.org/abs/hep-lat/9810051v1

Nucl.Phys.B547:281-302,1999

Physics

High Energy Physics

High Energy Physics - Lattice

30 pages in REVTeX

Scientific paper

10.1016/S0550-3213(99)00109-1

Discretizations of the Laplacian operator on non-hypercubical lattices are discussed in a systematic approach. It is shown that order $a^2$ errors always exist for discretizations involving only nearest neighbors. Among all lattices with the same density of lattice sites, the hypercubical lattices always have errors smaller than other lattices with the same number of spacetime dimensions. On the other hand, the four dimensional checkerboard lattice (also known as the Celmaster lattice) is the only lattice which is isotropic at order $a^2$. Explicit forms of the discretized Laplacian operators on root lattices are presented, and different ways of eliminating order $a^2$ errors are discussed.

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