We present the convergence analysis of the cell average technique, introduced in [J. Kumar et al., Powder Technol. 179 (2007) 205–228.], to solve the nonlinear continuous Smoluchowski coagulation equation. It is shown that the technique is second order accurate on uniform grids and first order accurate on non-uniform smooth (geometric) grids. As an essential ingredient, the consistency of the technique is thoroughly discussed.
DOI : 10.1051/m2an/2014035
Mots clés : Particles, coagulation, cell average technique, consistency, Lipschitz condition, convergence
@article{M2AN_2015__49_2_349_0,
author = {Giri, Ankik Kumar and Nagar, Atulya K.},
title = {Convergence of the cell average technique for {Smoluchowski} coagulation equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {349--372},
publisher = {EDP-Sciences},
volume = {49},
number = {2},
year = {2015},
doi = {10.1051/m2an/2014035},
zbl = {1315.65109},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2014035/}
}
TY - JOUR AU - Giri, Ankik Kumar AU - Nagar, Atulya K. TI - Convergence of the cell average technique for Smoluchowski coagulation equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 349 EP - 372 VL - 49 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014035/ DO - 10.1051/m2an/2014035 LA - en ID - M2AN_2015__49_2_349_0 ER -
%0 Journal Article %A Giri, Ankik Kumar %A Nagar, Atulya K. %T Convergence of the cell average technique for Smoluchowski coagulation equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 349-372 %V 49 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014035/ %R 10.1051/m2an/2014035 %G en %F M2AN_2015__49_2_349_0
Giri, Ankik Kumar; Nagar, Atulya K. Convergence of the cell average technique for Smoluchowski coagulation equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 2, pp. 349-372. doi : 10.1051/m2an/2014035. http://www.numdam.org/articles/10.1051/m2an/2014035/
and , Convergence of a finite volume scheme for coagulation-fragmentation equations. Math. Comput. 77 (2008) 851–882. | DOI | Zbl
P.B. Dubovskiǐ, Mathematical Theory of Coagulation. In vol. 23 of Lecture notes. Global Analysis Research Center, Seoul National university (1994). | Zbl
, and , Exact solutions for the coagulation-fragmentation equations. J. Phys. A: Math. Gen. 25 (1992) 4737–4744. | DOI | Zbl
and , Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math. Meth. Appl. Sci. 19 (1996) 571–591. | DOI | Zbl
and , An efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena. SIAM J. Sci. Comput. 22 (2000) 802–821. | DOI | Zbl
and , Stochastic particle approximations for Smoluchowski’s coagulation equation. Ann. Appl. Probab. 11 (2001) 1137–1165. | DOI | Zbl
, , and , Gelation and mass conservation in coagulation-fragmentation models. J. Differ. Equ. 195 (2003) 143–174. | DOI | Zbl
, and , Spline method for solving continuous batch grinding and similarity equations. Comput. Chem. Eng. 21 (1997) 1433–1440. | DOI
and , Mass-conserving solutions and non-conservative approximation to the smoluchowski coagulation equation. Arch. Math. 83 (2004) 558–567. | DOI | Zbl
and , Numerical simulation of the Smoluchowski coagulation equation. SIAM J. Sci. Comput. 25 (2004) 2004–2028. | DOI | Zbl
and , Existence of self-similar solutions to Smoluchowski’s coagulation equation. Commun. Math. Phys. 256 (2005) 589–609. | DOI | Zbl
and , Well-posedness of Smoluchowski’s coagulation equation for a class of homogeneous kernels. J. Funct. Anal. 233 (2006) 351–379. | DOI | Zbl
A.K. Giri, Mathematical and numerical analysis for coagulation-fragmentation equations. Ph.D. Thesis. Otto-von-Guericke-University Magdeburg, Germany, 2010.
and , Convergence analysis of sectional methods for solving aggregation population balance equations: The fixed pivot technique. Nonlinear Anal. Real World Appl. 14 (2013) 2068–2090. | DOI | Zbl
, and , The continuous coagulation equation with multiple fragmentation. J. Math. Anal. Appl. 374 (2011) 71–87. | DOI | Zbl
, and , Weak solutions to the continuous coagulation equation with multiple fragmentation. Nonlin. Anal. 75 (2012) 2199–2208. | DOI | Zbl
and , Uniqueness for the continuous coagulation-fragmentation equation with strong fragmentation. Z. Angew. Math. Phys. 62 (2011) 1047–1063. | DOI | Zbl
W. Hundsdorfer and J.G. Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations. Springer-Verlag New York, USA, 1st edition (2003). | Zbl
, , and , An efficient numerical technique for solving population balance equation involving aggregation, breakage, growth and nucleation. Powder Technol. 179 (2007) 205–228.
, , , and , Improved accuracy and convergence of discretized population balance for aggregation: The cell average technique. Chem. Eng. Sci. 61 (2006) 3327–3342. | DOI
and , On the solution of population balance equations by discretization-I. A fixed pivot technique. Chem. Eng. Sci. 51 (1996) 1311–1332. | DOI
and , On the solution of population balance equations by discretization II. A moving pivot technique. Chem. Eng. Sci. 51 (1996) 1333–1342. | DOI
and , Convergence analysis of sectional methods for solving breakage population balance equations - I: The fixed pivot technique. Numer. Math. 111 (2008) 81–108. | DOI | Zbl
and , Convergence analysis of sectional methods for solving breakage population balance equations - II: The cell average technique. Numer. Math. 110 (2008) 539–559. | DOI | Zbl
, Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math. Meth. Appl. Sci. 27 (2004) 703–721. | DOI | Zbl
, On a class of continuous coagulation- fragmentation equations. J. Differ. Equ. 167 (2000) 245–274 | DOI | Zbl
and , From the discrete to the continuous coagulation-fragmentation equations. Proc. Roy. Soc. Edinburgh 132A (2002) 1219–1248. | DOI | Zbl
, On the validity of the coagulation equation and the nature of runaway growth. Icarus 143 (2000) 74–86. | DOI
, Convergence of a discretization method for integro-differential equations. Numer. Math. 25 (1975) 103–107. | DOI | Zbl
, and , A semigroup approach to fragmentation models. SIAM J. Math. Anal. 28 (1997) 1158–1172. | DOI | Zbl
, and , An existence and uniqueness result for a coagulation and multiple-fragmentation equation. SIAM J. Math. Anal. 28 (1997) 1173–1190. | DOI | Zbl
and , Finite-element methods for steady-state population balance equations. AICHE J. 44 (1998) 2258–2272. | DOI
and , Finite-element scheme for solution of the dynamic population balance equation. AICHE J. 49 (2003) 1127–1139. | DOI
, , , and , Solution of population balance equation using quadrature method of moments with an adjustable factor. Chem. Eng. Sci. 62 (2007) 5897–5911. | DOI
, A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels. Math. Meth. Appl. Sci. 11 (1989) 627–648. | DOI | Zbl
Cité par Sources :
