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February  2019, 12(1): 195-216. doi: 10.3934/krm.2019009

An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions

Alina Chertock 1,, , Alexander Kurganov 2,3, , Mária Lukáčová-Medvi${\rm{\check{d}}}$ová 4, and  Șeyma Nur Özcan 1,

1. 

Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA
2. 

Department of Mathematics, Southern University of Science and Technology of China, Shenzhen 518055, China
3. 

Mathematics Department, Tulane University, New Orleans, LA 70118, USA
4. 

Institute of Mathematics, University of Mainz, Staudingerweg 9, 55099 Mainz, Germany

* Corresponding author: Alina Chertock

Received  February 2018 Published  July 2018

In this paper, we study two-dimensional multiscale chemotaxis models based on a combination of the macroscopic evolution equation for chemoattractant and microscopic models for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator which describes the velocity change of the cells. The parabolic scaling yields a non-dimensional kinetic model with a small parameter, which represents the mean free path of the cells. We propose a new asymptotic preserving numerical scheme that reflects the convergence of the studied micro-macro model to its macroscopic counterpart-the Patlak-Keller-Segel system-in the singular limit. The method is based on the operator splitting strategy and a suitable combination of the higher-order implicit and explicit time discretizations. In particular, we use the so-called even-odd decoupling and approximate the stiff terms arising in the singular limit implicitly. We prove that the resulting scheme satisfies the asymptotic preserving property. More precisely, it yields a consistent approximation of the Patlak-Keller-Segel system as the mean-free path tends to 0. The derived asymptotic preserving method is used to get better insight to the blowup behavior of two-dimensional kinetic chemotaxis model.

Keywords: Chemotaxis, kinetic equations, multiscale models, asymptotic preserving (AP) methods, operator splitting.
Mathematics Subject Classification: Primary: 92C17, 35Q20, 76M45; Secondary: 76P99, 65M06, 65M70.
Citation: Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.  Google Scholar

[2]

A. Bollermann, S. Noelle and M. Lukáčová-Medviďová, Finite volume evolution Galerkin methods for the shallow water equations with dry beds, Commun. Comput. Phys., 10 (2011), 371-404.  doi: 10.4208/cicp.220210.020710a.  Google Scholar

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V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

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V. Calvez, B. Perthame and M. Sharifi Tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, in Stochastic Analysis and Partial Differential Equations, vol. 429 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, 45–62. doi: 10.1090/conm/429/08229.  Google Scholar

[6]

J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Modeling & Simulation, 11 (2013), 336-361.  doi: 10.1137/110851687.  Google Scholar

[7]

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.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

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A. Chertock and A. Kurganov, A positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205.  doi: 10.1007/s00211-008-0188-0.  Google Scholar

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A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models, 5 (2012), 51–95, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=6915. doi: 10.3934/krm.2012.5.51.  Google Scholar

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A. Crestetto, N. Crouseilles and M. Lemou, Asymptotic-preserving scheme based on a finite volume/particle-in-cell coupling for Boltzmann-BGK-like equations in the diffusion scaling, in Finite Volumes for Complex Applications. VII. Elliptic, Parabolic and Hyperbolic Problems, vol. 78 of Springer Proc. Math. Stat., Springer, Cham, 2014,827–835. doi: 10.1007/978-3-319-05591-6_83.  Google Scholar

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N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, Kinet. Relat. Models, 4 (2011), 441-477.  doi: 10.3934/krm.2011.4.441.  Google Scholar

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G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal., 51 (2013), 1064-1087.  doi: 10.1137/12087606X.  Google Scholar

[14]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[15]

F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, Journal of Mathematical Biology, 50 (2005), 189-207.  doi: 10.1007/s00285-004-0286-2.  Google Scholar

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H. Gajewski, K. Zacharias and K. Gröger, Global behaviour of a reaction-diffusion system modelling chemotaxis, Mathematische Nachrichten, 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.  Google Scholar

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S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

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S. Gottlieb, D. I. Ketcheson and C.-W. Shu, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/7498.  Google Scholar

[19]

S. Guisset, S. Brull, E. D'Humières and B. Dubroca, Asymptotic-preserving well-balanced scheme for the electronic $ M_1$ model in the diffusive limit: particular cases, ESAIM Math. Model. Numer. Anal., 51 (2017), 1805-1826.  doi: 10.1051/m2an/2016079.  Google Scholar

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M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.   Google Scholar

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M. A. Herrero, E. Medina and J. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  doi: 10.1088/0951-7715/10/6/016.  Google Scholar

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M. A. Herrero and J. J. Velázquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 177-194.  doi: 10.1007/s002850050049.  Google Scholar

[23]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math., 61 (2000), 751-775 (electronic).  doi: 10.1137/S0036139999358167.  Google Scholar

[24]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[25]

T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125-144.  doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

[26]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[27]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences ⅰ, Jahresber. DMV, 105 (2003), 103-165.   Google Scholar

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D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences ⅱ, Jahresber. DMV, 106 (2004), 51-69.   Google Scholar

[29]

J. Hu, S. Jin and L. Wang, An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: a splitting approach, Kinet. Relat. Models, 8 (2015), 707-723.  doi: 10.3934/krm.2015.8.707.  Google Scholar

[30]

J. Hu, Q. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy, J. Sci. Comput., 62 (2015), 555-574.  doi: 10.1007/s10915-014-9869-2.  Google Scholar

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H. J. Hwang, K. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.  Google Scholar

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H. J. Hwang, K. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement, SIAM Journal on Mathematical Analysis, 36 (2005), 1177-1199.  doi: 10.1137/S0036141003431888.  Google Scholar

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W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of the American Mathematical Society, 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[34]

J. Jang, F. Li, J.-M. Qiu and T. Xiong, Analysis of asymptotic preserving DG-IMEX schemes for linear kinetic transport equations in a diffusive scaling, SIAM J. Numer. Anal., 52 (2014), 2048-2072.  doi: 10.1137/130938955.  Google Scholar

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S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Mat. Univ. Parma, 3 (2012), 177-216.   Google Scholar

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S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal., 35 (1998), 2405-2439 (electronic).  doi: 10.1137/S0036142997315962.  Google Scholar

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E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

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M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), 334-368.  doi: 10.1137/07069479X.  Google Scholar

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show all references

References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.  Google Scholar

[2]

A. Bollermann, S. Noelle and M. Lukáčová-Medviďová, Finite volume evolution Galerkin methods for the shallow water equations with dry beds, Commun. Comput. Phys., 10 (2011), 371-404.  doi: 10.4208/cicp.220210.020710a.  Google Scholar

[3]

N. Bournaveas and V. Calvez, Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 26, Elsevier, 2009, 1871–1895. doi: 10.1016/j.anihpc.2009.02.001.  Google Scholar

[4]

V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.  Google Scholar

[5]

V. Calvez, B. Perthame and M. Sharifi Tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, in Stochastic Analysis and Partial Differential Equations, vol. 429 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, 45–62. doi: 10.1090/conm/429/08229.  Google Scholar

[6]

J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Modeling & Simulation, 11 (2013), 336-361.  doi: 10.1137/110851687.  Google Scholar

[7]

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.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

[8]

A. Chertock and A. Kurganov, A positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205.  doi: 10.1007/s00211-008-0188-0.  Google Scholar

[9]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models, 5 (2012), 51–95, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=6915. doi: 10.3934/krm.2012.5.51.  Google Scholar

[10]

S. Childress and J. Percus, Nonlinear aspects of chemotaxis, Math. Biosc., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[11]

A. Crestetto, N. Crouseilles and M. Lemou, Asymptotic-preserving scheme based on a finite volume/particle-in-cell coupling for Boltzmann-BGK-like equations in the diffusion scaling, in Finite Volumes for Complex Applications. VII. Elliptic, Parabolic and Hyperbolic Problems, vol. 78 of Springer Proc. Math. Stat., Springer, Cham, 2014,827–835. doi: 10.1007/978-3-319-05591-6_83.  Google Scholar

[12]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, Kinet. Relat. Models, 4 (2011), 441-477.  doi: 10.3934/krm.2011.4.441.  Google Scholar

[13]

G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal., 51 (2013), 1064-1087.  doi: 10.1137/12087606X.  Google Scholar

[14]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[15]

F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, Journal of Mathematical Biology, 50 (2005), 189-207.  doi: 10.1007/s00285-004-0286-2.  Google Scholar

[16]

H. Gajewski, K. Zacharias and K. Gröger, Global behaviour of a reaction-diffusion system modelling chemotaxis, Mathematische Nachrichten, 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.  Google Scholar

[17]

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

[18]

S. Gottlieb, D. I. Ketcheson and C.-W. Shu, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/7498.  Google Scholar

[19]

S. Guisset, S. Brull, E. D'Humières and B. Dubroca, Asymptotic-preserving well-balanced scheme for the electronic $ M_1$ model in the diffusive limit: particular cases, ESAIM Math. Model. Numer. Anal., 51 (2017), 1805-1826.  doi: 10.1051/m2an/2016079.  Google Scholar

[20]

M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.   Google Scholar

[21]

M. A. Herrero, E. Medina and J. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  doi: 10.1088/0951-7715/10/6/016.  Google Scholar

[22]

M. A. Herrero and J. J. Velázquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 177-194.  doi: 10.1007/s002850050049.  Google Scholar

[23]

T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math., 61 (2000), 751-775 (electronic).  doi: 10.1137/S0036139999358167.  Google Scholar

[24]

T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[25]

T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125-144.  doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

[26]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[27]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences ⅰ, Jahresber. DMV, 105 (2003), 103-165.   Google Scholar

[28]

D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences ⅱ, Jahresber. DMV, 106 (2004), 51-69.   Google Scholar

[29]

J. Hu, S. Jin and L. Wang, An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: a splitting approach, Kinet. Relat. Models, 8 (2015), 707-723.  doi: 10.3934/krm.2015.8.707.  Google Scholar

[30]

J. Hu, Q. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy, J. Sci. Comput., 62 (2015), 555-574.  doi: 10.1007/s10915-014-9869-2.  Google Scholar

[31]

H. J. Hwang, K. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.  Google Scholar

[32]

H. J. Hwang, K. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement, SIAM Journal on Mathematical Analysis, 36 (2005), 1177-1199.  doi: 10.1137/S0036141003431888.  Google Scholar

[33]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of the American Mathematical Society, 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[34]

J. Jang, F. Li, J.-M. Qiu and T. Xiong, Analysis of asymptotic preserving DG-IMEX schemes for linear kinetic transport equations in a diffusive scaling, SIAM J. Numer. Anal., 52 (2014), 2048-2072.  doi: 10.1137/130938955.  Google Scholar

[35]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Mat. Univ. Parma, 3 (2012), 177-216.   Google Scholar

[36]

S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal., 35 (1998), 2405-2439 (electronic).  doi: 10.1137/S0036142997315962.  Google Scholar

[37]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM Journal on Scientific Computing, 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.  Google Scholar

[38]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[39]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[40]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[41]

A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations, SIAM J. Sci. Comput., 19 (1998), 2032-2050.  doi: 10.1137/S1064827595286177.  Google Scholar

[42]

A. Kurganov and M. Lukáčová-Medviďová, Numerical study of two-species chemotaxis models, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 131-152.  doi: 10.3934/dcdsb.2014.19.131.  Google Scholar

[43]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), 334-368.  doi: 10.1137/07069479X.  Google Scholar

[44]

G. I. Marchuk, Metody Rasshchepleniya, (Russian) [Splitting Methods] "Nauka", Moscow, 1988.  Google Scholar

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$||\rho||_\infty/M$ $ \verb"in time for varying values of"$ $M$; $N_x = N_y = 128$">Figure 1.  $ \verb"Example 1: Behavior of"$ $||\rho||_\infty/M$ $ \verb"in time for varying values of"$ $M$; $N_x = N_y = 128$
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$||\rho||_\infty$ $ \verb"in time for"$ $M = 7$ $ \verb"(left) and"$ $M = 9$ $ \verb"(right) on three consecutive meshes"$">Figure 2.  $ \verb"Example 1: Behavior of"$ $||\rho||_\infty$ $ \verb"in time for"$ $M = 7$ $ \verb"(left) and"$ $M = 9$ $ \verb"(right) on three consecutive meshes"$
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$||\rho||_\infty$ $ \verb"in time for"$ $M = 1$ $ \verb"(left)"$, $M = 8$ $ \verb"(middle) and"$ $M = 11$ $ \verb"(right) on four consecutive meshes"$">Figure 3.  $ \verb"Example 2a: Behavior of"$ $||\rho||_\infty$ $ \verb"in time for"$ $M = 1$ $ \verb"(left)"$, $M = 8$ $ \verb"(middle) and"$ $M = 11$ $ \verb"(right) on four consecutive meshes"$
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$\rho(x,y,T = 0.0005)$ $ \verb"for"$ $M = 11$ $ \verb"computed on the meshes with:"$ $N_x = N_y = 128$ $ \verb"(left) and"$ $N_x = N_y = 256$ $ \verb"(right)"$">Figure 4.  $ \verb"Example 2a: The density"$ $\rho(x,y,T = 0.0005)$ $ \verb"for"$ $M = 11$ $ \verb"computed on the meshes with:"$ $N_x = N_y = 128$ $ \verb"(left) and"$ $N_x = N_y = 256$ $ \verb"(right)"$
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$||\rho||_\infty$ $ \verb"in time for"$ $M = 8$ $ \verb"(left)"$, $M = 9.5$ $ \verb"(middle) and"$ $M = 11$ $ \verb"(right) on three consecutive meshes"$">Figure 5.  $ \verb"Example 2b: Behavior of"$ $||\rho||_\infty$ $ \verb"in time for"$ $M = 8$ $ \verb"(left)"$, $M = 9.5$ $ \verb"(middle) and"$ $M = 11$ $ \verb"(right) on three consecutive meshes"$
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$M = 3$">Figure 6.  $ \verb"Example 3: The displacement of the density for"$ $M = 3$
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$M = 7$">Figure 7.  $ \verb"Example 3: The displacement of the density for"$ $M = 7$
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$M = 11$">Figure 8.  $ \verb"Example 3: The displacement of the density for"$ $M = 11$
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Table 1.  $ \verb"Example 2b:"$ $L^\infty$- $ \verb"errors for"$ $M = 8, 9.5$ $ \verb"and"$ $11$ $ \verb"(from left to right)"$
$M=8$ $M=9.5$ $M=11$
$N$ $||e_{2N}||_\infty$ rate$_{4N}$ $||e_{2N}||_\infty$ rate$_{4N}$ $||e_{2N}||_\infty$ rate$_{4N}$
32 1.5680E-02 - 2.9056E-02 - 4.9613E-02 -
64 3.2150E-03 2.2860 8.1861E-03 1.8276 1.6752E-02 1.5663
128 8.4486E-04 1.9280 2.1204E-03 1.9488 4.5867E-03 1.8688
256 2.0985E-04 2.0093 5.3892E-04 1.9761 1.1662E-03 1.9756
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$M=8$ $M=9.5$ $M=11$
$N$ $||e_{2N}||_\infty$ rate$_{4N}$ $||e_{2N}||_\infty$ rate$_{4N}$ $||e_{2N}||_\infty$ rate$_{4N}$
32 1.5680E-02 - 2.9056E-02 - 4.9613E-02 -
64 3.2150E-03 2.2860 8.1861E-03 1.8276 1.6752E-02 1.5663
128 8.4486E-04 1.9280 2.1204E-03 1.9488 4.5867E-03 1.8688
256 2.0985E-04 2.0093 5.3892E-04 1.9761 1.1662E-03 1.9756
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