A numerical scheme for non-local aggregation with non-linear diffusion and approximations of social potential

Aggregations abound in nature, from cell formations to locust swarms. One method of modelling these aggregations is the non-local aggregation equation with the addition of degenerate diffusion. In this article we develop a finite volume based numerical scheme for this style of equation and perform an error, computation time, and convergence analysis. In addition we investigate two methods for approximating the non-local component. References A. J. Bernoff and C. M. Topaz. Nonlocal aggregation models: A primer of swarm equilibria. SIAM Rev. 55.4 (2013), pp. 709–747. doi: 10.1137/130925669 R. Bürger, D. Inzunza, P. Mulet, and L. M. Villada. Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour. Appl. Numer. Math. 144 (2019), pp. 234–252. doi: 10.1016/j.apnum.2019.04.018 J. A. Carrillo, A. Chertock, and Y. Huang. A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. In: Commun. Comput. Phys. 17.1 (2015), pp. 233–258. doi: 10.4208/cicp.160214.010814a J. R. Dormand and P. J. Prince. A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6.1 (1980), pp. 19–26. doi: 10.1016/0771-050X(80)90013-3 J. von zur Gathen and J. Gerhard. Modern computer algebra. 3rd ed. Cambridge University Press, 2013. doi: 10.1017/CBO9781139856065 F. Georgiou, J. Buhl, J. E. F. Green, B. Lamichhane, and N. Thamwattana. Modelling locust foraging: How and why food affects group formation. PLOS Comput. Biol. 17.7 (2021), e1008353. doi: 10.1371/journal.pcbi.1008353 F. Georgiou, B. P. Lamichhane, and N. Thamwattana. An adaptive numerical scheme for a partial integro-differential equation. Proceedings of the 18th Biennial Computational Techniques and Applications Conference, CTAC-2018. Ed. by B. Lamichhane, T. Tran, and J. Bunder. Vol. 60. ANZIAM J. 2019, pp. C187–C200. doi: 10.21914/anziamj.v60i0.14066 F. Georgiou, N. Thamwattana, and B. P. Lamichhane. Modelling cell aggregation using a modified swarm model. Proceedings of the 23rd International Congress on Modelling and Simulation, MODSIM2019. Vol. 6. 2019, pp. 22–27. doi: 10.36334/modsim.2019.a1.georgiou J. E. F. Green, S. L. Waters, J. P. Whiteley, L. Edelstein-Keshet, K. M. Shakesheff, and H. M. Byrne. Non-local models for the formation of hepatocyte–stellate cell aggregates. J. Theor. Bio. 267.1 (2010), pp. 106–120. doi: 10.1016/j.jtbi.2010.08.013 R. J. LeVeque. Finite-volume methods for hyperbolic Pproblems. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253 C. F. Van Loan. Introduction to Scientific Computing: A Matrix Vector Approach Using MATLAB. 1997. url: https://www.pearson.com/us/higher-education/program/Van- Loan-Introduction-to-Scientific-Computing-A-Matrix-Vector- Approach-Using-MATLAB-2nd-Edition/PGM215520.html A. Mogilner and L. Edelstein-Keshet. A non-local model for a swarm. J. Math. Bio. 38.6 (1999), pp. 534–570. doi: 10.1007/s002850050158 C. M. Topaz, A. L. Bertozzi, and M. A. Lewis. A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68 (2006), p. 1601. doi: 10.1007/s11538-006-9088-6 C. M. Topaz, M. R. D’Orsogna, L. Edelstein-Keshet, and A. J. Bernoff. Locust dynamics: Behavioral phase change and swarming. PLOS Comput. Bio. 8.8 (2012), e1002642. doi: 10.1371/journal.pcbi.100264