Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations

Rafael Bailo (1) , José A. Carrillo (2) , Hideki Murakawa (3) , Markus Schmidtchen (4)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 MI - Mathematical Institute [Oxford]
3 Ryukoku University
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

We study an implicit finite-volume scheme for non-linear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced in [3]. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

Mots clés: Finite-volume methods; convergence of numerical methods; drift-diffusion equations; integro-differential equations

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Rafael Bailo, José A. Carrillo, Hideki Murakawa, Markus Schmidtchen. Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations. Mathematical Models and Methods in Applied Sciences, 2020, 30 (13), pp.2487-2522. ⟨10.1142/S0218202520500487⟩. ⟨hal-03125839⟩