Numerical simulation of a kinetic model for chemotaxis

Kinetic and Related Models

Volumn 3, Issue 3, 2010, Pages 501-528

  Vauchelet, Nicolas ^a,b,c  

^a SORBONNE UNIVERSITÉ   (France)

^b CNRS   (France)

^c INRIA ROCQUENCOURT   (France)

Author keywords

Chemotaxis; Convergence analysis; Kinetic equations; Semi Lagrangian method

Indexed keywords

EID: 84860919966     PISSN: 19375093     EISSN: 19375077     Source Type: Journal    
DOI: 10.3934/krm.2010.3.501     Document Type: Article

References (41)

1. W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.
2. R. Belaouar, N. Crouseilles, P. Degond and E. Sonnendrücker, An asymptotically stable semi-Lagrangian scheme in the quasi-neutral limit, J. Sci. Comput., 41 (2009), 341-365.
3. N. Besse, Convergence of a semi-Lagrangian scheme for the one dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 42 (2004), 350-382.
4. C. K. Birsdall and A. B. Langdon, "Plasma Physics via Computer Simulation," McGraw-Hill, New York, 1985.
5. N. Bournaveas, A. Buguin, V. Calvez, B. Perthame, J. Saragosti and P. Silberzan, Mathe-matical description of bacterial traveling pulses, arXiv:0912.1792v1.
6. N. Bournaveas and V. Calvez, Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data, arXiv::0901.0503.
7. N. Bournaveas and V. Calvez, Global existence for the kinetic chemotaxis model without pointwise memory effects, and including internal variables, Kinet. Relat. Models, 1 (2008), 29-48.
8. N. Bournaveas, V. Calvez, S. Gutiérrez and B. Perthame, Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates, Comm. P.D.E., 33 (2008), 79-95.
9. J. P. Boris and D. L. Book, Solution of continuity equations by the method of flux-corrected transport, J. Comput. Phys., 20 (1976), 397-431.
10. C. J. Budd, R. Carretero-Gonźalez and R. D. Russell, Precise computations of chemotactic collapse using moving mesh methods, J. Comput. Phys., 202 (2005), 463-487.
11. F. A. C. C. Chalub, P. A. Markowich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.
12. C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), 330-351.
13. P. G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis, Finite Element Methods (Part 1), Vol. II, P. G. Ciarlet and J. L. Lions, eds., North-Holland, New-York, 1991, 17-351.
14. N. Crouseilles and F. Filbet, Numerical approximation of collisional plasma by high order methods, J. Comp. Phys., 201 (2004), 546-572.
15. N. Crouseilles, G. Latu and E. Sonnendrücker, Hermite spline interpolation on patches for parallely solving the Vlasov-Poisson equation, Int. J. Appl. Math. Comput. Sc., 17 (2007), 101-115.
16. Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol., 51 (2005), 595-615.
17. R. Erban and H. J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis, Disc. Cont. Dyn. Systems-Series B, 6 (2006), 1239-1260.
18. R. Erban and H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004/05), 361-391
19. R. Erban and H. G. Othmer, Taxis equations for amoeboid cells, J. Math. Biol., 54 (2007), 847-885.
20. F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.
21. F. Filbet, Ph. Laureņcot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.
22. F. Filbet, E. Sonnendrücker and P. Bertrand, Conservative numerical schemes for the Vlasov equation, J. Comput. Phys., 172 (2001), 166-187.
23. F. Filbet and E. Sonnendrücker, Comparison of Eulerian solver, Comput. Phys. Commun., 151 (2003), 247-266.
24. J. Haskovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151.
25. T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
26. D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its con-sequences, Jahrsber. Dtsch. Math.-Ver., 105 (2003), 103-165.
27. H. J. Hwang, K. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement, SIAM J. Math. Anal., 36 (2005), 1177-1199.
28. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as instability, J. Theor. Biol., 26 (1970), 399-415.
29. A. Marrocco, Numerical simulation of chemotaxis bacteria aggregation via mixed finite ele-ments, ESAIM M2AN, 37 (2003), 617-630.
30. N. Mittal, E. O. Budrene, M. P. Brenner and A. van Oudenaarden, Motility of Escherichia coli cells in clusters formed by chemotactic aggregation, PNAS, 100 (2003), 13259-13263.
31. H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.
32. H. G. Othmer and T. Hillen, The diffuion limit of transport equations. II. Chemotaxis equa-tions, SIAM J. Appl. Math., 62 (2002), 1222-1250.
33. H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
34. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 263-298.
35. B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.
36. F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equa-tion with non-smooth coefficients, Comm. Partial Differential Equations, 22 (1997), 337-358.
37. N. Saito, Conservative upwind finite element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Num. Analysis, 27 (2007), 332-365.
38. N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Applied Mathematics and Computation, 171 (2005), 72-90.
39. E. Sonnendrücker, J. Roche, P. Bertrand and A. Ghizzo, The semi-Lagrangian method for the numerical resolution of the Vlasov equation, J. Comput. Phys., 149 (1999), 201-220
40. E. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately inter-acting stochastic many particle systems, SIAM J. Appl. Math., 61 (2000), 183-212.
41. M. J. Tindall, S. L. Porter, P. K. Maini, G. Gaglia and J. P. Armitage, Overview of mathe-matical approaches used to model bacterial chemotaxis. I. The single cell, Bull. Math. Biol., 70 (2008), 1525-1569