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December  2016, 21(10): 3463-3478. doi: 10.3934/dcdsb.2016107

Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations

Hui Huang 1, and  Jian-Guo Liu 2,

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing, 100084
2. 

Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708

Received  December 2015 Revised  September 2016 Published  November 2016

In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is $C\Delta t$, where $\Delta t$ is the time-step size. Secondly, we prove the convergence rate $C\Delta t^2$ for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate $C\Delta t$.
Keywords: positivity preserving., chemotaxis, random particle method, Newtonian aggregation.
Mathematics Subject Classification: Primary: 65M12, 65M15; Secondary: 92C1.
Citation: Hui Huang, Jian-Guo Liu. Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3463-3478. doi: 10.3934/dcdsb.2016107
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show all references

References:
[1]

Mathematics of Computation, 37 (1981), 243-259. doi: 10.1090/S0025-5718-1981-0628693-0.  Google Scholar

[2]

Large-Scale Scientific Computing, Volume 2907 of the series Lecture Notes in Computer Science, (2014), 295-302. doi: 10.1007/978-3-540-24588-9_33.  Google Scholar

[3]

$2^{nd}$ edition, American Mathematical Society, 2010. doi: 10.1090/gsm/019.  Google Scholar

[4]

Applied Numerical Mathematics, 42 (2002), 159-176. doi: 10.1016/S0168-9274(01)00148-9.  Google Scholar

[5]

Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[6]

Communications on Pure and Applied Mathematics, 40 (1987), 189-220. doi: 10.1002/cpa.3160400204.  Google Scholar

[7]

H. Huang and J.-G. Liu, Error estimate of a random particle blob method for the Keller-Segel equation,, Mathematics of Computation, ().   Google Scholar

[8]

Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

Journal of Computational Physics, 171 (2001), 373-393. doi: 10.1006/jcph.2001.6802.  Google Scholar

[10]

$2^{nd}$ edition,American Mathematical Society, 2001. doi: 10.1090/gsm/014.  Google Scholar

[11]

J.-G. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations,, Mathematics of Computation, ().   Google Scholar

[12]

Cambridge University Press, 2002. doi: 10.1017/CBO9780511613203.  Google Scholar

[13]

Acta Numerica, 11 (2002), 341-434. doi: 10.1017/S0962492902000053.  Google Scholar

[14]

Atmospheric Environment, 35 (2001), 5749-5764. doi: 10.1016/S1352-2310(01)00368-5.  Google Scholar

[15]

Springer, New York, 2007. doi: 10.1007/978-3-7643-7842-4.  Google Scholar

[16]

$3^{rd}$ edition,Oxford University Press, 1985. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[17]

SIAM Journal on Numerical Analysis, 5 (1968), 506-517. doi: 10.1137/0705041.  Google Scholar

[18]

Springer, New York, 2011. doi: 10.1007/978-1-4419-7052-7.  Google Scholar

[19]

Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.  Google Scholar

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