Research Article High-Accuracy Approximation of High-Rank Derivatives: Isotropic Finite Differences Based on Lattice-Boltzmann Stencils Keijo Kalervo Mattila, 1 Luiz Adolfo Hegele Júnior, 2 and Paulo Cesar Philippi 1 1 Laboratory of Porous Media and hermophysical Properties, Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florian´ opolis, SC, Brazil 2 Department of Petroleum Engineering, State University of Santa Catarina, 88330-668 Balne´ ario Cambori´ u, SC, Brazil Correspondence should be addressed to Keijo Kalervo Mattila; keijo.mattila@lmpt.ufsc.br Received 8 August 2013; Accepted 12 October 2013; Published 29 January 2014 Academic Editors: J. Bana´ s and C. Yiu Copyright © 2014 Keijo Kalervo Mattila et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose isotropic inite diferences for high-accuracy approximation of high-rank derivatives. hese inite diferences are based on direct application of lattice-Boltzmann stencils. he presented inite-diference 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 veriication of the presented inite diferences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils. 1. Introduction he approximation of derivatives by inite diferences is the cornerstone of numerical computing. Forward, backward, and central diferences, the ive-point stencil for approximat- ing Laplacian in a two-dimensional domain, and the numer- ical analysis of the convergence rate of the related approx- imation errors, based on the application of Taylor series, require no introduction for anyone working in the ield of scientiic computing. Construction of inite-diference stencils for the approx- imation of high-rank derivatives in two or three dimen- sions, say gradient of Laplacian or biLaplacian, will already be a more advanced topic—even if achieved by solving a modest linear system of equations. A further complication is introduced when requiring an isotropic approximation of derivatives. More speciically, when the leading-order error term of the inite diference approximation is required to be an isotropic expression or, in other words, to be free of directional bias. Such a property may be essential, for example, when solving certain partial diferential equations. Conventional inite diferences are not isotropic in the above sense. Isotropic inite diferences of second-order accu- racy for the approximation of irst and second derivatives, both in two and three dimensions, together with a systematic procedure for constructing the diferences, are presented in [1]. Patra and Karttunen proceed further: they present up to fourth-order accurate isotropic stencils, in two and three dimensions, for the numerical computation of second, third, and fourth derivatives [2]. In the context of lattice Boltzmann methods, isotropic inite diferences have been well known for some time, mainly because of their importance in the approximation of interparticle forces in multiphase and multicomponent models. For example, in the so-called Shan-Chen multiphase model [3], as was remarked by Yuan and Schaefer [4], the originally proposed approximation of interparticle forces includes an isotropic inite-diference approximation of the gradient of the interaction potential. In fact, when the standard D2Q9 lattice-Boltzmann stencil is used, the approx- imation is equivalent to the one proposed by Kumar [1]. Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 142907, 16 pages http://dx.doi.org/10.1155/2014/142907