title: High-accuracy approximation of high-rank derivatives: isotropic finite differences based on lattice-Boltzmann stencils.
abstract: We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils.
authors.affiliation: Laboratory of Porous Media and Thermophysical Properties, Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil.
authors.forename: Keijo Kalervo
authors.initials: KK
authors.lastname: Mattila
authors.affiliation: Department of Petroleum Engineering, State University of Santa Catarina, 88330-668 Balneário Camboriú, SC, Brazil.
authors.forename: Luiz Adolfo
authors.initials: LA
authors.lastname: Hegele Júnior
authors.affiliation: Laboratory of Porous Media and Thermophysical Properties, Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil.
authors.forename: Paulo Cesar
authors.initials: PC
authors.lastname: Philippi
mesh_terms: D000465:Algorithms; D016880:Anisotropy; D003198:Computer Simulation; D020342:Finite Element Analysis; D008962:Models, Theoretical