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Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions
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 |
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. |
[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. |
[3] |
L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (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.
doi: 10.1016/S0168-9274(01)00148-9. |
[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. |
[6] |
J. Goodman, Convergence of the random vortex method,, Communications on Pure and Applied Mathematics, 40 (1987), 189.
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, (). 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. |
[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. |
[10] |
E. H. Lieb and M. Loss, Analysis,, $2^{nd}$ edition, (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, (). Google Scholar |
[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.
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.
doi: 10.1016/S1352-2310(01)00368-5. |
[15] |
B. Perthame, Transport Equations in Biology,, Springer, (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, (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.
doi: 10.1137/0705041. |
[18] |
M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations,, Springer, (2011).
doi: 10.1007/978-1-4419-7052-7. |
[19] |
M. Taylor, Partial Differential Equations III: Nonlinear Equations,, Springer, (2011).
doi: 10.1007/978-1-4419-7049-7. |
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. |
[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. |
[3] |
L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (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.
doi: 10.1016/S0168-9274(01)00148-9. |
[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. |
[6] |
J. Goodman, Convergence of the random vortex method,, Communications on Pure and Applied Mathematics, 40 (1987), 189.
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, (). 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. |
[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. |
[10] |
E. H. Lieb and M. Loss, Analysis,, $2^{nd}$ edition, (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, (). Google Scholar |
[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.
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.
doi: 10.1016/S1352-2310(01)00368-5. |
[15] |
B. Perthame, Transport Equations in Biology,, Springer, (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, (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.
doi: 10.1137/0705041. |
[18] |
M. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations,, Springer, (2011).
doi: 10.1007/978-1-4419-7052-7. |
[19] |
M. Taylor, Partial Differential Equations III: Nonlinear Equations,, Springer, (2011).
doi: 10.1007/978-1-4419-7049-7. |
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