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On the iota-delta function: a link between cellular automata and partial differential equations for modeling advection–dispersion from a constant source

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Abstract

Describing complex phenomena by means of cellular automata (CAs) has shown to be a very effective approach in pure and applied sciences. Most of the applications, however, rely on multidimensional CAs. For example, lattice gas CAs and lattice Boltzmann methods are widely used to simulate fluid flow and both share features with two-dimensional CAs. One-dimensional CAs, on the other hand, seem to have been neglected for modeling physical phenomena. In the present paper, we demonstrate that some one-dimensional CAs are equivalent to a stable linear finite difference scheme used to solve advection–diffusion partial differential equations (PDEs) by relying on the so-called iota-delta representation. Consequently, this work shows an important link between continuous and discrete models in general, and PDEs and CAs more in particular.

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Authors and Affiliations

  1. Department of Civil Engineering, University of Brasilia, Brasília, 70910-900, Brazil

    Luan Carlos de S. M. Ozelim

  2. Department of Civil and Environmental Engineering, University of Brasilia, Brasília, 70910-900, Brazil

    André Luís B. Cavalcante

  3. KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, 9000, Ghent, Belgium

    Jan M. Baetens

Authors
  1. Luan Carlos de S. M. Ozelim
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  2. André Luís B. Cavalcante
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  3. Jan M. Baetens
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Correspondence to Luan Carlos de S. M. Ozelim.

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Ozelim, L.C.d.S.M., Cavalcante, A.L.B. & Baetens, J.M. On the iota-delta function: a link between cellular automata and partial differential equations for modeling advection–dispersion from a constant source. J Supercomput 73, 700–712 (2017). https://doi.org/10.1007/s11227-016-1795-7

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  • DOI: https://doi.org/10.1007/s11227-016-1795-7

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