An Asymptotic Preserving Scheme for Capturing Concentrations in Age-structured Models Arising in Adaptive Dynamics - Inria - Institut national de recherche en sciences et technologies du numérique
Journal Articles Journal of Computational Physics Year : 2022

An Asymptotic Preserving Scheme for Capturing Concentrations in Age-structured Models Arising in Adaptive Dynamics

Abstract

We propose an asymptotic preserving (A-P) scheme for a population model structured by age and a phenotypical trait with or without mutations. As proved in [26], Dirac concentrations on particular phenotypical traits appear in the case without mutations, which makes the numerical resolution of the problem challenging. Inspired by its asymptotic behaviour, we apply a proper Wentzel-Kramers-Brillouin (WKB) representation of the solution to derive an A-P scheme, with which we can accurately capture the concentrations on a coarse, ε-independent mesh. The scheme is thoroughly analysed and important properties, including the A-P property, are rigorously proved. Furthermore, we observe nearly spectral accuracy in time in our numerical simulations. Next, we generalize the A-P scheme to the case with mutations, where a nonlinear Hamilton-Jacobi equation will be involved in the limiting model as ε → 0. It can be formally shown that the generalized scheme is A-P as well, and numerical experiments indicate that we can still accurately solve the problem on a coarse, ε-independent mesh in the phenotype space.
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Dates and versions

hal-02438316 , version 1 (14-01-2020)
hal-02438316 , version 2 (19-06-2022)

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Luís Almeida, Benoît Perthame, Xinran Ruan. An Asymptotic Preserving Scheme for Capturing Concentrations in Age-structured Models Arising in Adaptive Dynamics. Journal of Computational Physics, 2022, 464, pp.111335. ⟨10.1016/j.jcp.2022.111335⟩. ⟨hal-02438316v2⟩
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