American Institute of Mathematical Sciences

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

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

 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 $C\Delta t$.
Citation: Hui Huang, Jian-Guo Liu. Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations. Discrete and 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. [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. [3] L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American Mathematical Society, 2010. doi: 10.1090/gsm/019. [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. [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. [6] J. Goodman, Convergence of the random vortex method, Communications on Pure and Applied Mathematics, 40 (1987), 189-220. doi: 10.1002/cpa.3160400204. [7] H. Huang and J.-G. Liu, Error estimate of a random particle blob method for the Keller-Segel equation,, Mathematics of Computation, (). [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. [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. [10] E. H. Lieb and M. Loss, Analysis, $2^{nd}$ edition,American Mathematical Society, 2001. doi: 10.1090/gsm/014. [11] J.-G. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations,, Mathematics of Computation, (). [12] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613203. [13] R. I. McLachlan, G. Quispel and W. Reinout, Splitting methods, Acta Numerica, 11 (2002), 341-434. doi: 10.1017/S0962492902000053. [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. [15] B. Perthame, Transport Equations in Biology, Springer, New York, 2007. doi: 10.1007/978-3-7643-7842-4. [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. [17] G. Strang, On the construction and comparison of difference schemes, SIAM Journal on Numerical Analysis, 5 (1968), 506-517. doi: 10.1137/0705041. [18] M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, Springer, New York, 2011. doi: 10.1007/978-1-4419-7052-7. [19] M. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

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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. [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. [3] L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American Mathematical Society, 2010. doi: 10.1090/gsm/019. [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. [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. [6] J. Goodman, Convergence of the random vortex method, Communications on Pure and Applied Mathematics, 40 (1987), 189-220. doi: 10.1002/cpa.3160400204. [7] H. Huang and J.-G. Liu, Error estimate of a random particle blob method for the Keller-Segel equation,, Mathematics of Computation, (). [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. [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. [10] E. H. Lieb and M. Loss, Analysis, $2^{nd}$ edition,American Mathematical Society, 2001. doi: 10.1090/gsm/014. [11] J.-G. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations,, Mathematics of Computation, (). [12] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613203. [13] R. I. McLachlan, G. Quispel and W. Reinout, Splitting methods, Acta Numerica, 11 (2002), 341-434. doi: 10.1017/S0962492902000053. [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. [15] B. Perthame, Transport Equations in Biology, Springer, New York, 2007. doi: 10.1007/978-3-7643-7842-4. [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. [17] G. Strang, On the construction and comparison of difference schemes, SIAM Journal on Numerical Analysis, 5 (1968), 506-517. doi: 10.1137/0705041. [18] M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, Springer, New York, 2011. doi: 10.1007/978-1-4419-7052-7. [19] M. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.
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