American Institute of Mathematical Sciences

Advanced
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), 243-259. 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, 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]

L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American Mathematical Society, 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), 159-176. 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, Berlin, 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), 189-220. 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]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. 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), 373-393. doi: 10.1006/jcph.2001.6802.  Google Scholar

[10]

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

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), 341-434. 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), 5749-5764. doi: 10.1016/S1352-2310(01)00368-5.  Google Scholar

[15]

B. Perthame, Transport Equations in Biology, Springer, New York, 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,Oxford University Press, 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), 506-517. doi: 10.1137/0705041.  Google Scholar

[18]

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

[19]

M. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer, New York, 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), 243-259. 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, 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]

L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American Mathematical Society, 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), 159-176. 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, Berlin, 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), 189-220. 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]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. 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), 373-393. doi: 10.1006/jcph.2001.6802.  Google Scholar

[10]

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

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), 341-434. 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), 5749-5764. doi: 10.1016/S1352-2310(01)00368-5.  Google Scholar

[15]

B. Perthame, Transport Equations in Biology, Springer, New York, 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,Oxford University Press, 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), 506-517. doi: 10.1137/0705041.  Google Scholar

[18]

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

[19]

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

[1]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623
[2]

Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153
[3]

Hafedh Bousbih. Global weak solutions for a coupled chemotaxis non-Newtonian fluid. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 907-929. doi: 10.3934/dcdsb.2018212
[4]

Xin Xu. Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4327-4348. doi: 10.3934/cpaa.2020195
[5]

Jisheng Kou, Huangxin Chen, Xiuhua Wang, Shuyu Sun. A linear, decoupled and positivity-preserving numerical scheme for an epidemic model with advection and diffusion. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021094
[6]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[7]

Alan Mackey, Theodore Kolokolnikov, Andrea L. Bertozzi. Two-species particle aggregation and stability of co-dimension one solutions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1411-1436. doi: 10.3934/dcdsb.2014.19.1411
[8]

Laurent Desvillettes, Michèle Grillot, Philippe Grillot, Simona Mancini. Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4675-4692. doi: 10.3934/dcds.2018205
[9]

José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic & Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025
[10]

Yuri Kozitsky, Krzysztof Pilorz. Random jumps and coalescence in the continuum: Evolution of states of an infinite particle system. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 725-752. doi: 10.3934/dcds.2020059
[11]

Daniel Peterseim. Robustness of finite element simulations in densely packed random particle composites. Networks & Heterogeneous Media, 2012, 7 (1) : 113-126. doi: 10.3934/nhm.2012.7.113
[12]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503
[13]

Yanfei Wang, Qinghua Ma. A gradient method for regularizing retrieval of aerosol particle size distribution function. Journal of Industrial & Management Optimization, 2009, 5 (1) : 115-126. doi: 10.3934/jimo.2009.5.115
[14]

Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021097
[15]

Raffaele D’Ambrosio, Giuseppe De Martino, Beatrice Paternoster. A symmetric nearly preserving general linear method for Hamiltonian problems. Conference Publications, 2015, 2015 (special) : 330-339. doi: 10.3934/proc.2015.0330
[16]

Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1783-1799. doi: 10.3934/jimo.2019029
[17]

Anouar El Harrak, Amal Bergam, Tri Nguyen-Huu, Pierre Auger, Rachid Mchich. Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2163-2181. doi: 10.3934/dcdss.2021055
[18]

Harish Garg, Dimple Rani. Multi-criteria decision making method based on Bonferroni mean aggregation operators of complex intuitionistic fuzzy numbers. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2279-2306. doi: 10.3934/jimo.2020069
[19]

Frederic Heihoff. Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4703-4719. doi: 10.3934/dcdsb.2020120
[20]

Mihaela Negreanu, J. Ignacio Tello. On a comparison method to reaction-diffusion systems and its applications to chemotaxis. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2669-2688. doi: 10.3934/dcdsb.2013.18.2669

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences