### Metadata

### Abstract

A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation driven by a nonlocal nonlinear velocity field with low regularity. In particular, we allow the interacting potential to be pointy, in which case the velocity field may have discontinuities. Based on recent results of existence and uniqueness of a Filippov flow for this type of equations, we study an upwind finite volume numerical scheme and we prove that it is convergent at order $1/2$ in Wasserstein distance. The paper is illustrated by numerical simulations that indicate that this convergence order should be optimal.

### References

[AC84] Differential inclusions. Set-valued maps and viability theory, Grundlehren der Mathematischen Wissenschaften, Volume 264, Springer, 1984 | Zbl

[AGS05] Gradient flows in metric space of probability measures, Lectures in Mathematics, Birkhäuser, 2005 | Zbl

[BCDFP15] Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM, Control Optim. Calc. Var., Volume 21 (2015) no. 2, pp. 414-441 | DOI | MR | Zbl

[BCP97] A kinetic equation for granular media, RAIRO, Modélisation Math. Anal. Numér., Volume 31 (1997), pp. 615-641 | DOI | Numdam | MR | Zbl

[BG11] An estimate on the flow generated by monotone operators, Commun. Partial Differ. Equations, Volume 36 (2011) no. 4-6, pp. 777-796 | MR | Zbl

[BGL12] Characterization of radially symmetric finite time blowup in multidimensional aggregation equations, SIAM J. Math. Anal., Volume 44 (2012) no. 2, pp. 651-681 | DOI | MR | Zbl

[BGP05] Error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation, SIAM J. Numer. Anal., Volume 43 (2005) no. 2, pp. 578-603 | DOI | MR | Zbl

[BJ98] One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., Theory Methods Appl., Volume 32 (1998) no. 7, pp. 891-933 | DOI | MR | Zbl

[BL19] One-dimensional empirical measures, order statistics, and Kantorovich transport distances, Memoirs of the American Mathematical Society, Volume 1259, American Mathematical Society, 2019 | Zbl

[BLR11] ${L}^{p}$ theory for the multidimensional aggregation equation, Commun. Pure Appl. Math., Volume 64 (2011) no. 1, pp. 45-83 | DOI | MR | Zbl

[BV06] An integro-differential equation arising as a limit of individual cell-based models, J. Differ. Equations, Volume 222 (2006) no. 2, pp. 341-380 | DOI | MR | Zbl

[CB16] A blob method for the aggregation equation, Math. Comput., Volume 85 (2016) no. 300, pp. 1681-1717 | DOI | MR | Zbl

[CCH15] A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., Volume 17 (2015) no. 1, pp. 233-258 | DOI | MR | Zbl

[CDF + 11] Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., Volume 156 (2011) no. 2, pp. 229-271 | DOI | MR | Zbl

[CGLM12] A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., Volume 22 (2012) no. 4, 1150023, 34 pages | MR | Zbl

[CJLV16] The Filippov characteristic flow for the aggregation equation with mildly singular potentials, J. Differ. Equations, Volume 260 (2016) no. 1, pp. 304-338 | DOI | MR | Zbl

[CLM13] Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow, NoDEA, Nonlinear Differ. Equ. Appl., Volume 20 (2013) no. 3, pp. 523-537 | DOI | MR | Zbl

[CMV06] Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., Volume 179 (2006) no. 2, pp. 217-263 | DOI | MR | Zbl

[CPCCC15] Convergence of a linearly transformed particle method for aggregation equations (2015) (https://hal.archives-ouvertes.fr/hal-01180687) | Zbl

[Des04] An explicit a priori estimate for a finite volume approximation of linear advection on non-Cartesian grids, SIAM J. Numer. Anal., Volume 42 (2004) no. 2, pp. 484-504 | DOI | MR | Zbl

[DL11] Probabilistic analysis of the upwind scheme for transport equations, Arch. Ration. Mech. Anal., Volume 199 (2011) no. 1, pp. 229-268 | DOI | MR | Zbl

[DLV17] Analysis of finite volume upwind scheme for transport equation with discontinuous coefficients, J. Math. Pures Appl., Volume 108 (2017) no. 6, pp. 918-951 | DOI | Zbl

[Dob79] Vlasov equations, Funct. Anal. Appl., Volume 13 (1979), pp. 115-123 | DOI | Zbl

[DS05] Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., Volume 51 (2005) no. 6, pp. 595-615 | DOI | MR | Zbl

[Fil64] Differential equations with discontinuous right-hand side, Trans. Am. Math. Soc., Volume 42 (1964) no. 2, pp. 199-231 | DOI | Zbl

[FLP05] Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., Volume 50 (2005), pp. 189-207 | DOI | MR | Zbl

[GJ00] Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comput., Volume 69 (2000) no. 231, pp. 987-1015 | DOI | MR | Zbl

[Gol16] On the dynamics of large particle systems in the mean field limit, Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity (Lecture Notes in Applied Mathematics and Mechanics) Volume 3, Springer, 2016 | MR | Zbl

[GT06] Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM J. Numer. Anal., Volume 43 (2006), pp. 2590-2606 | DOI | MR | Zbl

[GV16] Numerical high-field limits in two-stream kinetic models and 1D aggregation equations, SIAM J. Sci. Comput., Volume 38 (2016) no. 1, p. A412-A434 | DOI | MR | Zbl

[HB10] Self-similar blowup solutions to an aggregation equation in ${\mathbb{R}}^{n}$, SIAM J. Appl. Math., Volume 70 (2010) no. 7, pp. 2582-2603 | DOI | MR | Zbl

[HB12] Asymptotics of blowup solutions for the aggregation equation, Discrete Contin. Dyn. Syst., Volume 17 (2012) no. 4, pp. 1309-1331 | MR | Zbl

[HLF94] Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comput., Volume 62 (1994) no. 206, pp. 497-530 | MR | Zbl

[JV13] Chemotaxis: from kinetic equations to aggregate dynamics, NoDEA, Nonlinear Differ. Equ. Appl., Volume 20 (2013), pp. 101-127 | DOI | MR | Zbl

[JV15] Numerical method for one-dimensional aggregation equations, SIAM J. Numer. Anal., Volume 53 (2015) no. 2, pp. 895-916 | DOI | MR | Zbl

[JV16] Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 3, pp. 1355-1382 | MR | Zbl

[KS70] Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., Volume 26 (1970) no. 3, pp. 399-415 | DOI | MR | Zbl

[Kuz76] The accuracy of some approximate methods for computing weak solutions of quasi-linear first order partial differential equation, Zh. Vychisl. Mat. Mat. Fiz., Volume 16 (1976), pp. 1489-1502 | Zbl

[LT04] Long time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal., Volume 172 (2004) no. 3, pp. 407-428 | MR | Zbl

[LV16] Analysis and simulation of nonlinear and nonlocal transport equations, Innovative algorithms and analysis (Springer INdAM Series) Volume 16, Springer, 2016, pp. 265-288 | DOI | Zbl

[MCO05] An interacting particle system modelling aggregation behavior: from individuals to populations, J. Math. Biol., Volume 50 (2005) no. 1, pp. 49-66 | DOI | MR | Zbl

[Mer07] ${L}^{\infty}$- and ${L}^{2}$-error estimates for a finite volume approximation of linear advection, SIAM J. Numer. Anal., Volume 46 (2007) no. 1, pp. 124-150 | DOI | MR | Zbl

[MV07] Error estimate for finite volume scheme, Numer. Math., Volume 106 (2007) no. 1, pp. 129-155 | DOI | MR | Zbl

[OL02] Diffusion and ecological problems: Modern perspectives, Interdisciplinary Applied Mathematics, Volume 14, Springer, 2002 | Zbl

[Pat53] Random walk with persistence and external bias, Bull. Math. Biophys., Volume 15 (1953), pp. 311-338 | DOI | MR | Zbl

[PR97] Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Commun. Partial Differ. Equations, Volume 22 (1997), pp. 337-358 | Zbl

[RR98] Mass Transportation Problems. Vol. I, Springer Series in Statistics, Volume 1998, Springer, 1998 | Zbl

[San15] Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling, Progress in Nonlinear Differential Equations and their Applications, Volume 87, Birkhäuser/Springer, 2015 | Zbl

[SS17] Convergence rates for upwind schemes with rough coefficients, SIAM J. Numer. Anal., Volume 55 (2017) no. 2, pp. 812-840 | DOI | MR | Zbl

[TB04] Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., Volume 65 (2004) no. 1, pp. 152-174 | DOI | MR | Zbl

[Tos04] Kinetic and hydrodynamic models of nearly elastic granular flows, Monatsh. Math., Volume 142 (2004) no. 1-2, pp. 179-192 | DOI | MR | Zbl

[Vil03] Topics in optimal transportation, Graduate Studies in Mathematics, Volume 58, American Mathematical Society, 2003 | MR | Zbl

[Vil09] Optimal transport, old and new, Grundlehren der Mathematischen Wissenschaften, Volume 338, Springer, 2009 | Zbl