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Lattice Boltzmann Method Analysis Tool (LBMAT)

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Abstract

A general computational tool for the derivation of equivalent partial differential equations (EPDEs) is presented for the lattice Boltzmann method (LBM) with general collision operators that include single relaxation time (SRT-LBM), multiple relaxation time (MRT-LBM), central LBM (CLBM), or cumulant LBM (CuLBM). The method can be used to recover the advection–diffusion equations (ADEs), Navier–Stokes equations (NSEs), and other problems that could be solved by LBM in all dimensions. The derivation of EPDEs starts with the discrete (lattice) Boltzmann equation for raw moments and uses spatio-temporal Taylor expansion of these moments to obtain a system of partial differential equations. Then, to recover the desired ADEs or NSEs with additional partial differential terms up to a given order, a computationally feasible algorithm is proposed to eliminate higher order moments. The algorithm for the derivation of EPDEs, under the name of LBMAT (Lattice Boltzmann Method Analysis Tool), is implemented in C++ using the GiNaC library for symbolic algebraic computations. In order to optimize memory demands for higher dimension LBM models such as D3Q27, a custom-tailored data structure for storing the terms of partial differential expressions is proposed. The implementation of LBMAT is available to the community as open-source software under the terms and conditions of the GNU general public license (GPL).

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Funding

The work was supported by the Ministry of Education, Youth and Sports of the Czech Republic under OP RDE grants no. CZ.02.1.01/0.0/0.0/16_019/0000765 and no. CZ.02.1.01/0.0/0.0/16_019/0000753, by the Ministry of Health of the Czech Republic project No. NV19-08-00071, by the Czech Science Foundation project no. 21-09093S, and by the National Science Center, Poland grant number UMO2018/31/B/ST8/00622.

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Authors and Affiliations

  1. Department of mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic

    Radek Fučík, Pavel Eichler, Jakub Klinkovský & Tomáš Oberhuber

  2. AGH University of Science and Technology, al. Mickiewicza 30, Krakow, 30-059, Poland

    Robert Straka

Authors
  1. Radek Fučík
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  2. Pavel Eichler
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  3. Jakub Klinkovský
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  4. Robert Straka
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  5. Tomáš Oberhuber
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Correspondence to Radek Fučík.

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The authors declare no competing interests.

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All data generated or analyzed during this study are included in this published article and its supplementary information files.

Code availability

Implementations of the algorithms used can be found at https://mmg-gitlab.fjfi.cvut.cz/gitlab/lbm/lbmat.

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Fučík, R., Eichler, P., Klinkovský, J. et al. Lattice Boltzmann Method Analysis Tool (LBMAT). Numer Algor 93, 1509–1525 (2023). https://doi.org/10.1007/s11075-022-01476-8

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  • DOI: https://doi.org/10.1007/s11075-022-01476-8

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