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

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  • 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 $C\Delta t$.
    Mathematics Subject Classification: Primary: 65M12, 65M15; Secondary: 92C17.


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  • [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.


    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.


    L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American Mathematical Society, 2010.doi: 10.1090/gsm/019.


    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.


    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.


    J. Goodman, Convergence of the random vortex method, Communications on Pure and Applied Mathematics, 40 (1987), 189-220.doi: 10.1002/cpa.3160400204.


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


    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.


    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.


    E. H. Lieb and M. Loss, Analysis, $2^{nd}$ edition,American Mathematical Society, 2001.doi: 10.1090/gsm/014.


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


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


    R. I. McLachlan, G. Quispel and W. Reinout, Splitting methods, Acta Numerica, 11 (2002), 341-434.doi: 10.1017/S0962492902000053.


    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.


    B. Perthame, Transport Equations in Biology, Springer, New York, 2007.doi: 10.1007/978-3-7643-7842-4.


    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.


    G. Strang, On the construction and comparison of difference schemes, SIAM Journal on Numerical Analysis, 5 (1968), 506-517.doi: 10.1137/0705041.


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


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

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