Article Contents
Article Contents

# Finite range method of approximation for balance laws in measure spaces

• * Corresponding author: Piotr Gwiazda
• In the following paper we reconsider a numerical scheme recently introduced in [10]. The method was designed for a wide class of size structured population models with a nonlocal term describing the birth process. Despite its numerous advantages it features the exponential growth in time of the number of particles constituting the numerical solution. We introduce a new algorithm free from this inconvenience. The improvement is based on the application the Finite Range Approximation to the nonlocal term. We prove the convergence of the derived method and provide the rate of its convergence. Moreover, the results are illustrated by numerical simulations applied to various test cases.

Mathematics Subject Classification: P92D25, 65M12, 65M75.

 Citation:

• Figure 1.  Function $f$ and its Finite Range Approximation

Figure 2.  Function $f^{\varepsilon}$ and its approximation $\bar f^{\varepsilon}$

Figure 3.  Order of convergence

Table 1.  The error of the FRA method for $\varepsilon=0.1$

 $\Delta t$ ${\rm{Err}}(1,\Delta t,0.1)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$ $1.0000\cdot 10^{-1}$ $0.0008865973$ - $5.0000\cdot 10^{-2}$ $0.0005886514$ 0.59086543 $2.5000\cdot 10^{-2}$ $0.0003434678$ 0.77723882 $1.2500\cdot 10^{-2}$ $0.0002668756$ 0.36400752 $6.2500\cdot 10^{-3}$ $0.0002400045$ 0.15310595 $3.1250\cdot 10^{-3}$ $0.0002258416$ 0.08775007 $1.5625\cdot 10^{-3}$ $0.0002187803$ 0.04582869 $7.8125\cdot 10^{-4}$ $0.0002154824$ 0.02191237

Table 2.  The error of the FRA method for $\varepsilon=0.0125$

 $\Delta t$ ${\rm{Err}}(1,\Delta t,0.0125)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}})/ \log 2$ $1.0000\cdot 10^{-1}$ $6.702793\cdot 10^{-4}$ - $5.0000\cdot 10^{-2}$ $4.255311\cdot 10^{-4}$ 0.6554979 $2.5000\cdot 10^{-2}$ $1.757573\cdot 10^{-4}$ 1.2756799 $1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 0.9135408 $6.2500\cdot 10^{-3}$ $5.658711\cdot 10^{-5}$ 0.7214983 $3.1250\cdot 10^{-3}$ $4.055238\cdot 10^{-5}$ 0.4806869 $1.5625\cdot 10^{-3}$ $3.332588\cdot 10^{-5}$ 0.2831438 $7.8125\cdot 10^{-4}$ $2.973577\cdot 10^{-5}$ 0.1644434

Table 3.  The error of the FRA method for $\varepsilon=0.0015625$

 $\Delta t$ ${\rm{Err}}(1,\Delta t,0.0015625)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$ $1.0000\cdot 10^{-1}$ $6.616854\cdot 10^{-4}$ - $5.0000\cdot 10^{-2}$ $4.139056\cdot 10^{-4}$ 0.6768439 $2.5000\cdot 10^{-2}$ $1.570582\cdot 10^{-4}$ 1.3980025 $1.2500\cdot 10^{-2}$ $7.418803\cdot 10^{-5}$ 1.0820407 $6.2500\cdot 10^{-3}$ $3.649949\cdot 10^{-5}$ 1.0820407 $3.1250\cdot 10^{-3}$ $1.883518\cdot 10^{-5}$ 0.9544464 $1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 0.8669200 $7.8125\cdot 10^{-4}$ $6.277910\cdot 10^{-6}$ 0.7181537

Table 4.  The error of the FRA method for $\varepsilon=\Delta t$

 $\Delta t$ ${\rm{Err}}(1,\Delta t,\Delta t)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,2\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$ $1.0000\cdot 10^{-1}$ $8.865973\cdot 10^{-4}$ - $5.0000\cdot 10^{-2}$ $4.797352\cdot 10^{-4}$ 0.8860407 $2.5000\cdot 10^{-2}$ $1.991736\cdot 10^{-4}$ 1.2682122 $1.2500\cdot 10^{-2}$ $9.330613\cdot 10^{-5}$ 1.0939823 $6.2500\cdot 10^{-3}$ $4.477800\cdot 10^{-5}$ 1.0591816 $3.1250\cdot 10^{-3}$ $2.162411\cdot 10^{-5}$ 1.0501496 $1.5625\cdot 10^{-3}$ $1.032763\cdot 10^{-5}$ 1.0661308 $7.8125\cdot 10^{-4}$ $4.763338\cdot 10^{-6}$ 1.1164650
•  L. M. Abia , O. Angulo  and  J. C. Lopez-Marcos , Numerical schemes for a size-structured cell population model with equal fission, Mathematical and Computer Modelling, 50 (2009) , 653-664.  doi: 10.1016/j.mcm.2009.05.023. A. S. Ackleh  and  K. Ito , Measure-valued solutions for a hierarchically size-structured population, Journal of Differential Equations, 217 (2005) , 431-455.  doi: 10.1016/j.jde.2004.12.013. A. S. Ackleh , B. G. Fitzpatrick  and  H. R. Thieme , Rate distributions and survival of the fittest: A formulation on the space of measures, Discrete and Continuous Dynamical Systems Series B, 5 (2005) , 917-928.  doi: 10.3934/dcdsb.2005.5.917. A. L. Bertozzi , T. Kolokonikov , H. Sun  and  D. Uminsky , Stability of ring patterns arising from 2d particle interactions, Physical Review E, 84 (2011) . C. K. Birdsal and A. B. Langdon, Plasma Physics Via Computer Simulation, McGraw-Hill, New York, 1985. A. Brannstrom , L. Carlsson  and  D. Simpson , On the convergence of the escalator boxcar train, SIAM J. Numer. Anal., 51 (2013) , 3213-3231.  doi: 10.1137/120893215. A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford Leture Series in Mathematics and its Applications vol. 20, Oxford University Press, 2000. J. A. Cañizo , J. A. Carrillo  and  J. Rosado , A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011) , 515-539.  doi: 10.1142/S0218202511005131. J. A. Carrillo , R. Colombo , P. Gwiazda  and  A. Ulikowska , Structured populations, cell growth and measure valued balance laws, Journal of Differential Equations, 252 (2012) , 3245-3277.  doi: 10.1016/j.jde.2011.11.003. J.A. Carrillo , P. Gwiazda  and  A. Ulikowska , Splitting particle methods for structured population models: Convergence and applications, Math. Models Methods Appl. Sci., 24 (2014) , 2171-2197.  doi: 10.1142/S0218202514500183. R. M. Colombo  and  G. Guerra , Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009) , 733-753.  doi: 10.3934/dcds.2009.23.733. G. H. Cottet  and  P. A. Raviart , Particle methods for the one-dimensional Vlasov-Poisson equations, SIAM J. Numer. Anal., 21 (1984) , 52-76.  doi: 10.1137/0721003. A. M. de Roos , Numerical methods for structured population models: The escalator boxcar train, Numerical Methods for Partial Differential Equations, 4 (1988) , 173-195.  doi: 10.1002/num.1690040303. A. M. de Roos and L. Persson, Population and Community Ecology of Ontogenetic Development, Monographs in Population Biology 51, Princeton University Press, Princeton, 2013. O. Diekmann  and  Ph. Getto , Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations, J. Differential Equations, 215 (2005) , 268-319.  doi: 10.1016/j.jde.2004.10.025. M. R. D'Orsogna, Y. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability and collapse, Phys. Rev. Lett., 96 (2006). J. Evers , S. Hille  and  A. Muntean , Mild solutions to a measure-valued mass evolution problem with flux boundary conditions, Journal of Differential Equations, 259 (2015) , 1068-1097.  doi: 10.1016/j.jde.2015.02.037. K. Ganguly  and  H. D. Victory Jr , On the convergence of particle methods for multidimensional Vlasov-Poisson systems, SIAM J. Numer. Anal., 26 (1989) , 249-288.  doi: 10.1137/0726015. J. Goodman , T. Y. Hou  and  J. Lowengrub , Convergence of the point vortex method for the 2-D Euler equations, Comm. Pure Appl. Math., 43 (1990) , 415-430.  doi: 10.1002/cpa.3160430305. P. Gwiazda , J. Jabƚoński , A. Marciniak-Czochra  and  A. Ulikowska , Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Numerical Methods for Partial Differential Equations, 30 (2014) , 1797-1820.  doi: 10.1002/num.21879. P. Gwiazda , G. Jamróz  and  A. Marciniak-Czochra , Models of discrete and continuous cell differentiation in the framework of transport equation, SIAM Journal on Mathematical Analysis, 44 (2012) , 1103-1133.  doi: 10.1137/11083294X. P. Gwiazda , T. Lorenz  and  A. Marciniak-Czochra , A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010) , 2703-2735.  doi: 10.1016/j.jde.2010.02.010. P. Gwiazda  and  A. Marciniak-Czochra , Structured population equations in metric spaces, Journal of Hyperbolic Differential Equations, 7 (2010) , 733-773.  doi: 10.1142/S021989161000227X. F. H. Harlow , The particle-in-cell computing method for fluid dynamics, Methods in computational physics, 3 (1964) , 319-343. D. Issautier , Convergence of a weighted particle method for solving the Boltzmann (BGK) equation, SIAM J. Numer. Anal., 33 (1996) , 2099-2119.  doi: 10.1137/S0036142994266856. B. Piccoli  and  F. Rossi , Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014) , 335-358.  doi: 10.1007/s00205-013-0669-x. B. Piccoli  and  A. Tosin , Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011) , 707-738.  doi: 10.1007/s00205-010-0366-y. P. A. Raviart, An analysis of particle methods, Numerical Methods in Fluid Dynamics, Lecture Notes in Math. , Springer, Berlin, 1127 (1985), 243-324. doi: 10.1007/BFb0074532. E. Tadmor , A review of numerical methods for nonlinear partial differential equations, Bulletin of the American Mathematical Society, 49 (2012) , 507-554.  doi: 10.1090/S0273-0979-2012-01379-4. C. Villani, Topics in Optimal Transportation, volume 58 of Graduate studies in mathematics, American Mathematical Society, 2003. doi: 10.1007/b12016. G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89. Marcel Dekker, Inc., New York, 1985. M. Westdickenberg  and  J. Wilkening , Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations, M2AN Math. Model. Numer. Anal., 44 (2010) , 133-166.  doi: 10.1051/m2an/2009043.

Figures(3)

Tables(4)