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Finite difference schemes for a structured population model in the space of measures

1 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA
2 Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (1428) Pabellón I-Ciudad Universitaria-Buenos Aires-Argentina

Special Issues: Mathematical Modeling with Measures

We present two finite-difference methods for approximating solutions to a structured population model in the space of non-negative Radon Measures. The first method is a first-order upwind-based scheme and the second is high-resolution method of second-order. We prove that the two schemes converge to the solution in the Bounded-Lipschitz norm. Several numerical examples demonstrating the order of convergence and behavior of the schemes around singularities are provided. In particular, these numerical results show that for smooth solutions the upwind and high-resolution methods provide a first-order and a second-order approximation, respectively. Furthermore, for singular solutions the second-order high-resolution method is superior to the first-order method.
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Keywords finite difference schemes; high-resolution methods; structured populations; non-negative Radon measures; bounded-Lipschitz norm

Citation: Azmy S. Ackleh, Rainey Lyons, Nicolas Saintier. Finite difference schemes for a structured population model in the space of measures. Mathematical Biosciences and Engineering, 2020, 17(1): 747-775. doi: 10.3934/mbe.2020039


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