Abstract
The paper is concerned with development of a new finite-volume method for a class of chemotaxis models and for a closely related haptotaxis model. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. The first step in the derivation of the new method is made by adding an equation for the chemoattractant concentration gradient to the original system. We then show that the convective part of the resulting system is typically of a mixed hyperbolic-elliptic type and therefore straightforward numerical methods for the studied system may be unstable. The proposed method is based on the application of the second-order central-upwind scheme, originally developed for hyperbolic systems of conservation laws in Kurganov et al. (SIAM J Sci Comput 21:707–740, 2001), to the extended system of PDEs. We show that the proposed second-order scheme is positivity preserving, which is a very important stability property of the method. The scheme is applied to a number of two-dimensional problems including the most commonly used Keller–Segel chemotaxis model and its modern extensions as well as to a haptotaxis system modeling tumor invasion into surrounding healthy tissue. Our numerical results demonstrate high accuracy, stability, and robustness of the proposed scheme.
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References
Adler J.: Chemotaxis in bacteria. Ann. Rev. Biochem. 44, 341–356 (1975)
Anderson A.R.A.: A hybrid mathematical model of solid tumor invasion: the importance of cell adhesion. Math. Med. Biol. IMA J. 22, 163–186 (2005)
Ascher U.M., Ruuth S.J., Spiteri R.J.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Special issue on time integration (Amsterdam, 1996). Appl. Numer. Math. 25, 151–167 (1997)
Ascher U.M., Ruuth S.J., Wetton B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32, 797–823 (1995)
Ayati B.P., Webb G.F., Anderson A.R.A.: Computational methods and results for structured multiscale models of tumor invasion. Multiscale Model. Simul. 5, 1–20 (2006)
Bonner J.T.: The Cellular Slime Molds, 2nd edn. Princeton University Press, Princeton (1967)
Budrene E.O., Berg H.C.: Complex patterns formed by motile cells of escherichia coli. Nature 349, 630–633 (1991)
Budrene E.O., Berg H.C.: Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376, 49–53 (1995)
Calvez V., Carrillo J.A.: Volume effects in the Keller–Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 86, 155–175 (2006)
Carter S.B.: Principles of cell motility: the direction of cell movement and cancer invasion. Nature 208, 1183–1187 (1965)
Carter S.B.: Haptotaxis and the mechanism of cell motility. Nature 213, 256–260 (1967)
Chertock, A., Kurganov, A., Petrova, G.: Fast explicit operator splitting method for convection-diffusion equations. Int. J. Numer. Methods Fluids (2008, in press)
Childress S., Percus J.K.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981)
Cohen M.H., Robertson A.: Wave propagation in the early stages of aggregation of cellular slime molds. J. Theor. Biol. 31, 101–118 (1971)
Filbet F.: A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. Numer. Math. 104, 457–488 (2006)
Godlewski E., Raviart P.-A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, New York (1996)
Godunov S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 47, 271–306 (1959)
Gottlieb S., Shu C.-W., Tadmor E.: High order time discretization methods with the strong stability property. SIAM Rev. 43, 89–112 (2001)
Herrero M.A., Velázquez J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. 24, 633–683 (1997)
Higueras, I., Roldán, T.: Positivity-preserving and entropy-decaying IMEX methods. In: Ninth International Conference Zaragoza-Pau on Applied Mathematics and Statistics. Monogr. Semin. Mat. Garcia Galdeano, vol. 33, pp. 129–136. Prensas University Zaragoza, Zaragoza (2006)
Hillen T., Painter K.J.: Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26, 280–301 (2001)
Horstmann D.: From 1970 until now: the Keller–Segel model in chemotaxis and its consequences I. Jahresber. DMV 105, 103–165 (2003)
Horstmann D.: From 1970 until now: the Keller–Segel model in chemotaxis and its consequences II. Jahresber. DMV 106, 51–69 (2004)
Hundsdorfer W., Verwer J.: Numerical solution of time-dependent advection-diffusion-reaction equations. Springer Series in Computational Mathematics, vol. 33. Springer, Berlin (2003)
Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Keller E.F., Segel L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)
Keller E.F., Segel L.A.: Treveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 235–248 (1971)
Kröner D.: Numerical Schemes for Conservation Laws. Wiley, Chichester (1997)
Kurganov, A.: Central-upwind schemes for balance laws. Application to the Broadwell model. In: Finite Volumes for Complex Applications, III (Porquerolles, 2002), pp. 351–358. Hermes Sci. Publ., Paris (2002)
Kurganov A., Levy D.: Central-upwind schemes for the Saint–Venant system. M2AN Math. Model. Numer. Anal. 36, 397–425 (2002)
Kurganov A., Lin C.-T.: On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2, 141–163 (2007)
Kurganov A., Noelle S., Petrova G.: Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21, 707–740 (2001)
Kurganov A., Petrova G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5, 133–160 (2007)
Kurganov A., Tadmor E.: New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 241–282 (2000)
van Leer B.: Towards the ultimate conservative difference scheme, V. a second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)
LeVeque R.: Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, London (2002)
Lie K.-A., Noelle S.: On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24, 1157–1174 (2003)
Lin C.-S., Ni W.-M., Takagi I.: Large amplitude stationary solutions to a chemotaxis system. J. Differ. Equ. 72, 1–27 (1988)
Manoussaki D.: A mechanochemical model of angiogenesis and vasculogenesis. M2AN Math. Model. Numer. Anal. 37, 581–599 (2003)
Marrocco A.: 2D simulation of chemotaxis bacteria aggregation. M2AN Math. Model. Numer. Anal. 37, 617–630 (2003)
Nanjundiah V.: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42, 63–105 (1973)
Nessyahu H., Tadmor E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)
Pareschi L., Russo G.: Implicit-Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25, 129–155 (2005)
Patlak C.S.: Random walk with persistence and external bias. Bull. Math: Biophys. 15, 311–338 (1953)
Perthame, B.: Transport equations in biology. Frontiers in Mathematics. Birkhäuser, Basel (2007)
Prescott L.M., Harley J.P., Klein D.A.: Microbiology, 3rd edn. Wm. C. Brown Publishers, Chicago (1996)
Samarskii A.A., Gulin A.V.: Ustoichivost raznostnykh skhem (Russian) [Stability of difference schemes], 2nd edn. Editorial URSS, Moscow (2004)
Saito N.: Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis. IMA J. Numer. Anal. 27, 332–365 (2007)
Sweby P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984)
Tyson R., Lubkin S.R., Murray J.D.: Model and analysis of chemotactic bacterial patterns in a liquid medium. J. Math. Biol. 38, 359–375 (1999)
Tyson R., Lubkin S.R., Murray J.D.: A minimal mechanism for bacterial pattern formation. Proc. Roy. Soc. Lond. B 266, 299–304 (1999)
Tyson R., Stern L.G., LeVeque R.J.: Fractional step methods applied to a chemotaxis model. J. Math. Biol. 41, 455–475 (2000)
Walker, C., Webb, G.F.: Global Existence of classical solutions for a haptotaxis model (preprint)
Woodward D., Tyson R., Myerscough M., Murray J., Budrene E., Berg H.: Spatio-temporal patterns generated by S. typhimurium. Biophys. J. 68, 2181–2189 (1995)
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Chertock, A., Kurganov, A. A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111, 169–205 (2008). https://doi.org/10.1007/s00211-008-0188-0
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DOI: https://doi.org/10.1007/s00211-008-0188-0