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 mutation. As proved in [24], Dirac concentrations on particular phenotypical traits appear in the case without mutation, which makes the numerical resolution of the problem challenging. Inspired by its asymptotic behaviour, we apply a proper 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 mutation, 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 resolve the problem on a coarse, ε-independent mesh in the phenotype space.
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Submitted on : Tuesday, January 14, 2020 - 10:55:16 AM
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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. 2020. ⟨hal-02438316⟩

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