Abstract
In this paper, we introduce and analyze an asymptotic-preserving scheme for Lotka–Volterra parabolic equations. It is a class of nonlinear and nonlocal stiff equations, which describes the evolution of a population structured with phenotypic trait. In a regime of large time scale and small mutations, the population concentrates at a set of dominant traits. The dynamics of this concentration is described by a constrained Hamilton–Jacobi equation, which is a system coupling a Hamilton–Jacobi equation with a Lagrange multiplier determined by a constraint. This coupling makes the equation nonlocal. Moreover, the constraint does not enjoy much regularity, since it can have jumps. The scheme we propose is convergent in all the regimes, and enjoys stability in the long time and small mutations limit. Moreover, we prove that the limiting scheme converges towards the viscosity solution of the constrained Hamilton–Jacobi equation, despite the lack of regularity of the constraint. The theoretical analysis of the schemes is illustrated and complemented with numerical simulations.
Original language | English |
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Pages (from-to) | 103-153 |
Number of pages | 51 |
Journal | Numerische Mathematik |
Volume | 154 |
Issue number | 1-2 |
DOIs | |
Publication status | Published - 2023 |
Bibliographical note
Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-careOtherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.