Isotropic discrete Laplacian operators from lattice hydrodynamics

TL;DR

This paper introduces a method for deriving isotropic discrete Laplacian operators using lattice hydrodynamics principles, resulting in efficient, isotropic stencils suitable for various computational fluid dynamics simulations.

Contribution

It presents a novel approach to constructing isotropic Laplacian operators from lattice hydrodynamics, enabling smaller, more efficient stencils with guaranteed isotropy.

Findings

Isotropic Laplacians can be derived from lattice hydrodynamics principles.

Small stencils with as few as 15 points in 3D achieve isotropy.

The method extends to higher-order discretizations and other differential operators.

Abstract

We show that discrete schemes developed for lattice hydrodynamics provide an elegant and physically transparent way of deriving Laplacians with isotropic discretisation error. Isotropy is guaranteed whenever the Laplacian weights follow from the discrete Maxwell-Boltzmann equilibrium since these are, by construction, isotropic on the lattice. We also point out that stencils using as few as 15 points in three dimensions, generate isotropic Laplacians. These computationally efficient Laplacians can be used in cell-dynamical and hybrid lattice Boltzmann simulations, in favor of popular anisotropic Laplacians, which make use of larger stencils. The method can be extended to provide discretisations of higher order and for other differential operators, such the gradient, divergence and curl.