Error estimate of a random particle blob method for the Keller-Segel equation

Huang, Hui; Liu, Jian-Guo

doi:10.1090/mcom/3174

Error estimate of a random particle blob method for the Keller-Segel equation
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by Hui Huang and Jian-Guo Liu;

Math. Comp. 86 (2017), 2719-2744

DOI: https://doi.org/10.1090/mcom/3174

Published electronically: February 15, 2017

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Abstract:

We establish an optimal error estimate for a random particle blob method for the Keller-Segel equation in $\mathbb {R}^d~(d\geq 2)$. With a blob size $\varepsilon =h^{\kappa }$ $(1/2<\kappa <1)$, we prove a rate $h|\ln h|$ of convergence in $\ell ^p_h$ $(p>\frac {d}{1-\kappa })$ norm up to a probability $1-h^{C|\ln h|}$, where $h$ is the initial grid size.

References

Christopher Anderson and Claude Greengard, On vortex methods, SIAM J. Numer. Anal. 22 (1985), no. 3, 413–440. MR 787568, DOI 10.1137/0722025
J. Thomas Beale and Andrew Majda, Vortex methods. I. Convergence in three dimensions, Math. Comp. 39 (1982), no. 159, 1–27. MR 658212, DOI 10.1090/S0025-5718-1982-0658212-5
J. Thomas Beale and Andrew Majda, Vortex methods. II. Higher order accuracy in two and three dimensions, Math. Comp. 39 (1982), no. 159, 29–52. MR 658213, DOI 10.1090/S0025-5718-1982-0658213-7
Shen Bian and Jian-Guo Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$, Comm. Math. Phys. 323 (2013), no. 3, 1017–1070. MR 3106502, DOI 10.1007/s00220-013-1777-z
Niklas Boers and Peter Pickl, On mean field limits for dynamical systems, J. Stat. Phys. 164 (2016), no. 1, 1–16. MR 3509045, DOI 10.1007/s10955-015-1351-5
Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, DOI 10.1017/S0022112073002016
Georges-Henri Cottet and Petros D. Koumoutsakos, Vortex methods, Cambridge University Press, Cambridge, 2000. Theory and practice. MR 1755095, DOI 10.1017/CBO9780511526442
Katy Craig and Andrea L. Bertozzi, A blob method for the aggregation equation, Math. Comp. 85 (2016), no. 300, 1681–1717. MR 3471104, DOI 10.1090/mcom3033
David Freedman, Brownian motion and diffusion, 2nd ed., Springer-Verlag, New York-Berlin, 1983. MR 686607
A. Friedman, Stochastic Differential Equations and Applications, volume 1, Academic Press, 2014.
Jonathan Goodman, Convergence of the random vortex method, Comm. Pure Appl. Math. 40 (1987), no. 2, 189–220. MR 872384, DOI 10.1002/cpa.3160400204
Carl Graham and Denis Talay, Stochastic simulation and Monte Carlo methods, Stochastic Modelling and Applied Probability, vol. 68, Springer, Heidelberg, 2013. Mathematical foundations of stochastic simulation. MR 3097957, DOI 10.1007/978-3-642-39363-1
Ole H. Hald, Convergence of vortex methods for Euler’s equations. II, SIAM J. Numer. Anal. 16 (1979), no. 5, 726–755. MR 543965, DOI 10.1137/0716055
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology 26 (1970), no. 3, 399–415.
Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
J.-G. Liu and R. Yang, Propagation of chaos for the keller-segel equation with a logarithmic cut-off, preprint.
Jian-Guo Liu and Rong Yang, A random particle blob method for the Keller-Segel equation and convergence analysis, Math. Comp. 86 (2017), no. 304, 725–745. MR 3584546, DOI 10.1090/mcom/3118
J.-G. Liu and Z. Yuan, Convergence of diffusion-drift many particle systems in probability under Sobolev norm, in P. Gonçalves and A. J. Soares, editors, From Particle Systems to Partial Differential Equations III, Springer, 2016.
Ding-Gwo Long, Convergence of the random vortex method in two dimensions, J. Amer. Math. Soc. 1 (1988), no. 4, 779–804. MR 958446, DOI 10.1090/S0894-0347-1988-0958446-1
C. Marchioro and M. Pulvirenti, Vortex methods in two-dimensional fluid dynamics, Lecture Notes in Physics, vol. 203, Springer-Verlag, Berlin, 1984. MR 750980
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
B. Oksendal, Stochastic differential equations: an introduction with applications, Springer Science & Business Media, 2013.
Hirofumi Osada, Propagation of chaos for the two-dimensional Navier-Stokes equation, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 1, 8–11. MR 839794
David Pollard, Convergence of stochastic processes, Springer Series in Statistics, Springer-Verlag, New York, 1984. MR 762984, DOI 10.1007/978-1-4612-5254-2
Michel Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224. MR 756174, DOI 10.1090/memo/0307