A convergent numerical scheme to a McKendrick–von Foerster equation with diffusion

Abstract

In this paper, a numerical scheme for a nonlinear McKendrick–von Foerster equation with diffusion in age with the Dirichlet boundary condition, and the Robin boundary condition are proposed. The main idea to derive the scheme is to use the discretization based on the method of characteristics to the convection part, and the finite difference method to the rest of the terms. The nonlocal terms are dealt with the quadrature methods. As a result, an implicit scheme is obtained for the boundary value problem under consideration. The consistency, and the convergence of the proposed numerical scheme is established.

Keywords:

• Age-structured population model
• finite difference method
• interpolation
• consistency
• convergence

2020 AMS Subject Classifications:

• 35K20
• 65N12
• 92D25

Acknowledgements

The authors would like to thank Prof. Jordi Ripoll for fruitful discussions. The authors are very grateful to the anonymous reviewers for their comments and suggestions that greatly helped to improve this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author would like to thank BITS Pilani, Hyderabad campus, for providing financial support under Research Initiation Grant  [grant number BITS/GAU/RIG/2019/H0712]. The second author would like to acknowledge the support of DST-SERB, India, under MATRICS [grant number MTR/2019/000848].