American Institute of Mathematical Sciences

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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
References:
[1]

J. T. Beale and A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations,, Mathematics of Computation, 37 (1981), 243.  doi: 10.1090/S0025-5718-1981-0628693-0.  Google Scholar

[2]

M. Botchev, I.Faragó and Á. Havasi, Testing weighted splitting schemes on a one-column transport-chemistry model,, Large-Scale Scientific Computing, (2014), 295.  doi: 10.1007/978-3-540-24588-9_33.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (2010).  doi: 10.1090/gsm/019.  Google Scholar

[4]

A. Gerisch and J. G. Verwer, Operator splitting and approximate factorization for taxis-diffusion-reaction models,, Applied Numerical Mathematics, 42 (2002), 159.  doi: 10.1016/S0168-9274(01)00148-9.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2001).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[6]

J. Goodman, Convergence of the random vortex method,, Communications on Pure and Applied Mathematics, 40 (1987), 189.  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]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, Journal of Theoretical Biology, 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

D. Lanser, J. G. Blom and J. G. Verwer, Time integration of the shallow water equations in spherical geometry,, Journal of Computational Physics, 171 (2001), 373.  doi: 10.1006/jcph.2001.6802.  Google Scholar

[10]

E. H. Lieb and M. Loss, Analysis,, $2^{nd}$ edition, (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]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511613203.  Google Scholar

[13]

R. I. McLachlan, G. Quispel and W. Reinout, Splitting methods,, Acta Numerica, 11 (2002), 341.  doi: 10.1017/S0962492902000053.  Google Scholar

[14]

F. Müller, Splitting error estimation for micro-physical-multiphase chemical systems in meso-scale air quality models,, Atmospheric Environment, 35 (2001), 5749.  doi: 10.1016/S1352-2310(01)00368-5.  Google Scholar

[15]

B. Perthame, Transport Equations in Biology,, Springer, (2007).  doi: 10.1007/978-3-7643-7842-4.  Google Scholar

[16]

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods,, $3^{rd}$ edition, (1985).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[17]

G. Strang, On the construction and comparison of difference schemes,, SIAM Journal on Numerical Analysis, 5 (1968), 506.  doi: 10.1137/0705041.  Google Scholar

[18]

M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations,, Springer, (2011).  doi: 10.1007/978-1-4419-7052-7.  Google Scholar

[19]

M. Taylor, Partial Differential Equations III: Nonlinear Equations,, Springer, (2011).  doi: 10.1007/978-1-4419-7049-7.  Google Scholar

show all references

References:
[1]

J. T. Beale and A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations,, Mathematics of Computation, 37 (1981), 243.  doi: 10.1090/S0025-5718-1981-0628693-0.  Google Scholar

[2]

M. Botchev, I.Faragó and Á. Havasi, Testing weighted splitting schemes on a one-column transport-chemistry model,, Large-Scale Scientific Computing, (2014), 295.  doi: 10.1007/978-3-540-24588-9_33.  Google Scholar

[3]

L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (2010).  doi: 10.1090/gsm/019.  Google Scholar

[4]

A. Gerisch and J. G. Verwer, Operator splitting and approximate factorization for taxis-diffusion-reaction models,, Applied Numerical Mathematics, 42 (2002), 159.  doi: 10.1016/S0168-9274(01)00148-9.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2001).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[6]

J. Goodman, Convergence of the random vortex method,, Communications on Pure and Applied Mathematics, 40 (1987), 189.  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]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, Journal of Theoretical Biology, 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

D. Lanser, J. G. Blom and J. G. Verwer, Time integration of the shallow water equations in spherical geometry,, Journal of Computational Physics, 171 (2001), 373.  doi: 10.1006/jcph.2001.6802.  Google Scholar

[10]

E. H. Lieb and M. Loss, Analysis,, $2^{nd}$ edition, (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]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511613203.  Google Scholar

[13]

R. I. McLachlan, G. Quispel and W. Reinout, Splitting methods,, Acta Numerica, 11 (2002), 341.  doi: 10.1017/S0962492902000053.  Google Scholar

[14]

F. Müller, Splitting error estimation for micro-physical-multiphase chemical systems in meso-scale air quality models,, Atmospheric Environment, 35 (2001), 5749.  doi: 10.1016/S1352-2310(01)00368-5.  Google Scholar

[15]

B. Perthame, Transport Equations in Biology,, Springer, (2007).  doi: 10.1007/978-3-7643-7842-4.  Google Scholar

[16]

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods,, $3^{rd}$ edition, (1985).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[17]

G. Strang, On the construction and comparison of difference schemes,, SIAM Journal on Numerical Analysis, 5 (1968), 506.  doi: 10.1137/0705041.  Google Scholar

[18]

M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations,, Springer, (2011).  doi: 10.1007/978-1-4419-7052-7.  Google Scholar

[19]

M. Taylor, Partial Differential Equations III: Nonlinear Equations,, Springer, (2011).  doi: 10.1007/978-1-4419-7049-7.  Google Scholar

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