A Mickens-type discretization of a diffusive model with nonpolynomial advection/convection and reaction terms

Authors

  • Jorge E. Macías-Díaz,

    Corresponding author
    1. Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes, Ags., Mexico
    • Correspondence to: Jorge E. Macías-Díaz, Departamento de Matemáticas y Física, Universidad Autóonoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, Ags. 20131, Mexico jemacias@correo.uaa.mx

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  • José Villa-Morales

    1. Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes, Ags., Mexico
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  • 2010 Mathematics Subject Classification. Primary 65D10, 65M06, 65Q30; Secondary 35K55, 92C45

Abstract

In this manuscript, we follow a nonlinear approach in the problem of numerically estimating solutions of a generic, diffusive partial differential equation with nonpolynomial advection/convection, and reaction laws. The nonlocal, finite-difference methodology presented in this manuscript has the advantage of conditionally preserving the properties of non-negativity, boundedness, and temporal and spatial monotonicity of approximations, which are characteristics that are inherent to some kink-like solutions of particular forms of the mathematical model under investigation. We establish theorems on the stability of the technique, on the existence and uniqueness of non-negative and bounded solutions of our computational model, and on the property of preservation of monotonicity. As corollaries, we show that the method is capable of preserving the spatial and temporal monotonicity of numerical approximations. We provide some illustrative, numerical simulations that support our analytical results. Our simulations show that the method provides good approximations to the exact solutions of the particular problems considered and that the properties of non-negativity, boundedness, and monotonicity (both temporal and spatial) are conditionally preserved at each iteration. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 652–669, 2015

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