• Previous Article
    Convergence of the compressible isentropic magnetohydrodynamic equations to the incompressible magnetohydrodynamic equations in critical spaces
  • KRM Home
  • This Issue
  • Next Article
    A review of the mean field limits for Vlasov equations
December  2014, 7(4): 713-737. doi: 10.3934/krm.2014.7.713

Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations

1. 

RICAM, Austrian Academy of Sciences, Altenberger Strasse 69, Linz, 4040, Austria

2. 

Department of Mathematics, IIT Kharagpur, Kharagpur, 721302, India

3. 

Institute for Analysis and Numerics, Otto-von-Guericke University Magdeburg, Universitatsplatz 2, Magdeburg, D-39106, Germany

Received  March 2014 Revised  August 2014 Published  November 2014

This paper presents stability and convergence analysis of a finite volume scheme for solving aggregation, breakage and the combined processes by showing consistency of the method and Lipschitz continuity of numerical fluxes. It is investigated that the finite volume scheme is second order convergent independently of the meshes for pure breakage problem while for pure aggregation and coupled problems, it indicates second order convergence on uniform and non-uniform smooth meshes. Furthermore, it gives only first order accuracy on non-uniform meshes. The mathematical results of convergence analysis are also demonstrated numerically for several test problems.
Citation: Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713
References:
[1]

J. P. Bourgade and F. Filbet, Convergence of a finite volume scheme for coagulation-fragmentation equations,, Mathematics of Computation, 77 (2008), 851.  doi: 10.1090/S0025-5718-07-02054-6.  Google Scholar

[2]

R. B. Diemer and J. H. Olson, A moment methodology for coagulation and breakage problems: Part 1-analytical solution of the steady-state population balance,, Chem. Eng. Sci., 57 (2002), 2193.   Google Scholar

[3]

P. B. Dubovskii, V. A. Galkin and I. W. Stewart, Exact solutions for the coagulation-fragmentation equations,, J. of Physics A: Mathematical and General, 25 (1992), 4737.  doi: 10.1088/0305-4470/25/18/009.  Google Scholar

[4]

M. Escobedo, P. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models,, Journal of Differential Equations, 195 (2003), 143.  doi: 10.1016/S0022-0396(03)00134-7.  Google Scholar

[5]

F. Filbet and P. Laurençot, Numerical simulation of the Smoluchowski coagulation equation,, SIAM Journal on Scientific Computing, 25 (2004), 2004.  doi: 10.1137/S1064827503429132.  Google Scholar

[6]

F. Filbet and P. Laurençot, Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation,, Archiv der Mathematik, 83 (2004), 558.  doi: 10.1007/s00013-004-1060-9.  Google Scholar

[7]

Y. P. Gokhale, R. Kumar, J. Kumar, G. Warnecke, J. Tomas and W. Hintz, Disintegration process of surface stabilized sol-gel $\text{TiO}_2$ nanoparticles by population balances,, Chem. Eng. Sci., 64 (2009), 5302.   Google Scholar

[8]

W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations,, $1^{st}$ edition, (2003).  doi: 10.1007/978-3-662-09017-6.  Google Scholar

[9]

J. Kumar, Numerical Approximations of Population Balance Equations in Particulate Systems,, Ph.D thesis, (2006).   Google Scholar

[10]

J. Kumar and G. Warnecke, Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique,, Numer. Math., 111 (2008), 81.  doi: 10.1007/s00211-008-0174-6.  Google Scholar

[11]

R. Kumar, J. Kumar and G. Warnecke, Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms,, Math. Models and Methods in App. Sc., 23 (2013), 1235.  doi: 10.1142/S0218202513500085.  Google Scholar

[12]

R. Kumar, Numerical Analysis of Finite Volume Schemes for Solving Population Balance Equations,, Ph.D thesis, (2011).   Google Scholar

[13]

S. Kumar and D. Ramkrishna, On the solution of population balance equations by discretization-I: a fixed pivot technique,, Chem. Eng. Sci., 51 (1996), 1311.  doi: 10.1016/0009-2509(96)88489-2.  Google Scholar

[14]

P. L. C. Lage, Comments on the "an analytical solution to the population balance equation with coalescence and breakage-the special case with constant number of particles" by D.P. Patil and J.R.G. Andrews,, Chem. Eng. Sci., 57 (2002), 4253.   Google Scholar

[15]

W. Lamb, Existence and uniqueness results for the continuous coagulation and fragmentation equation,, Mathematical Methods in the Applied Sciences, 27 (2004), 703.  doi: 10.1002/mma.496.  Google Scholar

[16]

K. Lee and T. Matsoukas, Simultaneous coagulation and breakage using constant-N Monte Carlo,, Powder Technology, 110 (2000), 82.   Google Scholar

[17]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, $1^{st}$ edition, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[18]

P. Linz, Convergence of a discretization method for integro-differential equations,, Numer. Math., 25 (1975), 103.  doi: 10.1007/BF01419532.  Google Scholar

[19]

J. Makino, T. Fukushige, Y. Funato and E. Kokubo, On the mass distribution of planetesimals in the early runaway stage,, New Astronomy, 3 (1998), 411.  doi: 10.1016/S1384-1076(98)00021-9.  Google Scholar

[20]

D. L. Marchisio and R. O. Fox, Solution of population balance equations using the direct quadrature method of moments,, Journal of Aerosol Science, 36 (2005), 43.  doi: 10.1016/j.jaerosci.2004.07.009.  Google Scholar

[21]

D. J. McLaughlin, W. Lamb and A. C. McBride, Existence and uniqueness results for the non-autonomous coagulation and multiple-fragmentation equation,, Mathematical Methods in the Applied Sciences, 21 (1998), 1067.  doi: 10.1002/(SICI)1099-1476(19980725)21:11<1067::AID-MMA985>3.0.CO;2-X.  Google Scholar

[22]

Z. A. Melzak, A scalar transport equation,, Trans. of the American Math. Society, 85 (1957), 547.  doi: 10.1090/S0002-9947-1957-0087880-6.  Google Scholar

[23]

S. Motz, A. Mitrovic and E. D. Gilles, Comparison of numerical methods for the simulation of dispersed phase systems,, Chem. Eng. Sci., 57 (2002), 4329.  doi: 10.1016/S0009-2509(02)00349-4.  Google Scholar

[24]

S. Qamar and G. Warnecke, Solving population balance equations for two-component aggregation by a finite volume scheme,, Chem. Eng. Sci., 62 (2007), 679.  doi: 10.1016/j.ces.2006.10.001.  Google Scholar

[25]

D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering,, $1^{st}$ edition, (2000).   Google Scholar

[26]

W. T. Scott, Analytic studies of cloud droplet coalescence,, J. of the Atmospheric Sci., 25 (1968), 54.  doi: 10.1175/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2.  Google Scholar

[27]

M. Sommer, F. Stenger, W. Peukert and N. J. Wagner, Agglomeration and breakage of nanoparticles in stirred media mills-a comparison of different methods and models,, Chem. Eng. Sci., 61 (2006), 135.  doi: 10.1016/j.ces.2004.12.057.  Google Scholar

[28]

M. Vanni, Approximate population balance equations for aggregation-breakage processes,, J. of Colloid and Interface Science, 221 (2002), 143.  doi: 10.1006/jcis.1999.6571.  Google Scholar

[29]

R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerization,, J. of Physics A: Mathematical and General, 18 (1985), 3027.  doi: 10.1088/0305-4470/18/15/026.  Google Scholar

[30]

R. M. Ziff, New solution to the fragmentation equation,, J. of Physics A: Mathematical and General, 24 (1991), 2821.  doi: 10.1088/0305-4470/24/12/020.  Google Scholar

show all references

References:
[1]

J. P. Bourgade and F. Filbet, Convergence of a finite volume scheme for coagulation-fragmentation equations,, Mathematics of Computation, 77 (2008), 851.  doi: 10.1090/S0025-5718-07-02054-6.  Google Scholar

[2]

R. B. Diemer and J. H. Olson, A moment methodology for coagulation and breakage problems: Part 1-analytical solution of the steady-state population balance,, Chem. Eng. Sci., 57 (2002), 2193.   Google Scholar

[3]

P. B. Dubovskii, V. A. Galkin and I. W. Stewart, Exact solutions for the coagulation-fragmentation equations,, J. of Physics A: Mathematical and General, 25 (1992), 4737.  doi: 10.1088/0305-4470/25/18/009.  Google Scholar

[4]

M. Escobedo, P. Laurençot, S. Mischler and B. Perthame, Gelation and mass conservation in coagulation-fragmentation models,, Journal of Differential Equations, 195 (2003), 143.  doi: 10.1016/S0022-0396(03)00134-7.  Google Scholar

[5]

F. Filbet and P. Laurençot, Numerical simulation of the Smoluchowski coagulation equation,, SIAM Journal on Scientific Computing, 25 (2004), 2004.  doi: 10.1137/S1064827503429132.  Google Scholar

[6]

F. Filbet and P. Laurençot, Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation,, Archiv der Mathematik, 83 (2004), 558.  doi: 10.1007/s00013-004-1060-9.  Google Scholar

[7]

Y. P. Gokhale, R. Kumar, J. Kumar, G. Warnecke, J. Tomas and W. Hintz, Disintegration process of surface stabilized sol-gel $\text{TiO}_2$ nanoparticles by population balances,, Chem. Eng. Sci., 64 (2009), 5302.   Google Scholar

[8]

W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations,, $1^{st}$ edition, (2003).  doi: 10.1007/978-3-662-09017-6.  Google Scholar

[9]

J. Kumar, Numerical Approximations of Population Balance Equations in Particulate Systems,, Ph.D thesis, (2006).   Google Scholar

[10]

J. Kumar and G. Warnecke, Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique,, Numer. Math., 111 (2008), 81.  doi: 10.1007/s00211-008-0174-6.  Google Scholar

[11]

R. Kumar, J. Kumar and G. Warnecke, Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms,, Math. Models and Methods in App. Sc., 23 (2013), 1235.  doi: 10.1142/S0218202513500085.  Google Scholar

[12]

R. Kumar, Numerical Analysis of Finite Volume Schemes for Solving Population Balance Equations,, Ph.D thesis, (2011).   Google Scholar

[13]

S. Kumar and D. Ramkrishna, On the solution of population balance equations by discretization-I: a fixed pivot technique,, Chem. Eng. Sci., 51 (1996), 1311.  doi: 10.1016/0009-2509(96)88489-2.  Google Scholar

[14]

P. L. C. Lage, Comments on the "an analytical solution to the population balance equation with coalescence and breakage-the special case with constant number of particles" by D.P. Patil and J.R.G. Andrews,, Chem. Eng. Sci., 57 (2002), 4253.   Google Scholar

[15]

W. Lamb, Existence and uniqueness results for the continuous coagulation and fragmentation equation,, Mathematical Methods in the Applied Sciences, 27 (2004), 703.  doi: 10.1002/mma.496.  Google Scholar

[16]

K. Lee and T. Matsoukas, Simultaneous coagulation and breakage using constant-N Monte Carlo,, Powder Technology, 110 (2000), 82.   Google Scholar

[17]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, $1^{st}$ edition, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[18]

P. Linz, Convergence of a discretization method for integro-differential equations,, Numer. Math., 25 (1975), 103.  doi: 10.1007/BF01419532.  Google Scholar

[19]

J. Makino, T. Fukushige, Y. Funato and E. Kokubo, On the mass distribution of planetesimals in the early runaway stage,, New Astronomy, 3 (1998), 411.  doi: 10.1016/S1384-1076(98)00021-9.  Google Scholar

[20]

D. L. Marchisio and R. O. Fox, Solution of population balance equations using the direct quadrature method of moments,, Journal of Aerosol Science, 36 (2005), 43.  doi: 10.1016/j.jaerosci.2004.07.009.  Google Scholar

[21]

D. J. McLaughlin, W. Lamb and A. C. McBride, Existence and uniqueness results for the non-autonomous coagulation and multiple-fragmentation equation,, Mathematical Methods in the Applied Sciences, 21 (1998), 1067.  doi: 10.1002/(SICI)1099-1476(19980725)21:11<1067::AID-MMA985>3.0.CO;2-X.  Google Scholar

[22]

Z. A. Melzak, A scalar transport equation,, Trans. of the American Math. Society, 85 (1957), 547.  doi: 10.1090/S0002-9947-1957-0087880-6.  Google Scholar

[23]

S. Motz, A. Mitrovic and E. D. Gilles, Comparison of numerical methods for the simulation of dispersed phase systems,, Chem. Eng. Sci., 57 (2002), 4329.  doi: 10.1016/S0009-2509(02)00349-4.  Google Scholar

[24]

S. Qamar and G. Warnecke, Solving population balance equations for two-component aggregation by a finite volume scheme,, Chem. Eng. Sci., 62 (2007), 679.  doi: 10.1016/j.ces.2006.10.001.  Google Scholar

[25]

D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering,, $1^{st}$ edition, (2000).   Google Scholar

[26]

W. T. Scott, Analytic studies of cloud droplet coalescence,, J. of the Atmospheric Sci., 25 (1968), 54.  doi: 10.1175/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2.  Google Scholar

[27]

M. Sommer, F. Stenger, W. Peukert and N. J. Wagner, Agglomeration and breakage of nanoparticles in stirred media mills-a comparison of different methods and models,, Chem. Eng. Sci., 61 (2006), 135.  doi: 10.1016/j.ces.2004.12.057.  Google Scholar

[28]

M. Vanni, Approximate population balance equations for aggregation-breakage processes,, J. of Colloid and Interface Science, 221 (2002), 143.  doi: 10.1006/jcis.1999.6571.  Google Scholar

[29]

R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerization,, J. of Physics A: Mathematical and General, 18 (1985), 3027.  doi: 10.1088/0305-4470/18/15/026.  Google Scholar

[30]

R. M. Ziff, New solution to the fragmentation equation,, J. of Physics A: Mathematical and General, 24 (1991), 2821.  doi: 10.1088/0305-4470/24/12/020.  Google Scholar

[1]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[2]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

[3]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[4]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[5]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[6]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[7]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[8]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270

[9]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[10]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[11]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[12]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304

[13]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

[14]

Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105

[15]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]