Jüngel, A., & Vetter, M. (2023). A Convergent Entropy-Dissipating BDF2 Finite-Volume Scheme for a Population Cross-Diffusion System. Computational Methods in Applied Mathematics, 24(3), 725–746. https://doi.org/10.1515/cmam-2023-0009
Cross-Diffusion Equations; Discrete Entropy Dissipation; Finite-Volume Method; Linear Multistep Method; Population Dynamics; Rao Entropy
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Abstract:
A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system case. Some numerical experiments in one and two space dimensions underline the theoretical results.
