This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford–Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate L1 space to a weak solution to the problem. By applying the methods and theory of operator semigroups, we are able to show that these weak solutions are unique and necessarily classical (differentiable) solutions, a degree of regularity not generally established when finite volume schemes are applied to such problems. Furthermore, this approach enabled us to derive a bound for the error induced by the truncation of the mass domain, and also establish the convergence of the truncated solutions as the truncation point is increased without bound. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence as the mesh is refined, whilst also verifying the bound on the truncation error.
Keywords: Fragmentation equations, finite volume scheme, mixed discrete-continuous fragmentation model
@article{M2AN_2021__55_3_1067_0,
author = {Baird, Graham and S\"uli, Endre},
title = {A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1067--1101},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2020088},
mrnumber = {4269462},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020088/}
}
TY - JOUR AU - Baird, Graham AU - Süli, Endre TI - A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1067 EP - 1101 VL - 55 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020088/ DO - 10.1051/m2an/2020088 LA - en ID - M2AN_2021__55_3_1067_0 ER -
%0 Journal Article %A Baird, Graham %A Süli, Endre %T A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1067-1101 %V 55 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020088/ %R 10.1051/m2an/2020088 %G en %F M2AN_2021__55_3_1067_0
Baird, Graham; Süli, Endre. A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 1067-1101. doi: 10.1051/m2an/2020088
[1] and , Convergence to equilibrium in a system of reacting polymers. Comm. Math. Phys. 65 (1979) 203–230. | MR | Zbl | DOI
[2] , , and , Vector-valued laplace transforms and Cauchy Problems. In: Monographs in Mathematics. Birkhauser Verlag (2001). | MR | Zbl
[3] , On a Monte Carlo approach to the Smoluchowski equations,. Monte Carlo Methods App. 5 (1999) 1–18. | MR | Zbl | DOI
[4] , Mixed discrete–continuous fragmentation equations. Ph.D. thesis, University of Oxford (2017). https://ora.ox.ac.uk/objects/uuid:311da0da-6801-4120-9129-d95786a153b6.
[5] and , A mixed discrete–continuous fragmentation model. J. Math. Anal. App. 473 (2019) 273–296. | MR | DOI
[6] , On an extension of Kato-Voigt perturbation theorem for substochastic semigroups and its application. Taiwanese J. Math. 5 (2001) 169–191. | MR | Zbl | DOI
[7] , Transport processes with coagulation and strong fragmentation. Discrete Contin. Dyn. Syst. B 17 (2012) 445–472. | MR | Zbl
[8] and , Perturbations of Positive Semigroups with Applications. Springer-Verlag, London (2006). | MR | Zbl
[9] and , On the application of substochastic semigroup theory to fragmentation models with mass loss. J. Math. Anal. Appl. 284 (2003) 9–30. | MR | Zbl | DOI
[10] and , Coagulation, fragmentation and growth processes in a size structured population. Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 563–585. | MR | Zbl
[11] and , Analytic fragmentation semigroups and continuous coagulation–fragmentation equations with unbounded rates. J. Math. Anal. App. 391 (2012) 312–322. | MR | Zbl | DOI
[12] and , The discrete fragmentation equations: semigroups, compactness and asynchronous exponential growth. Kinetic Related Models 5 (2012) 223–236. | MR | Zbl | DOI
[13] and , On the existence of moments of solutions to fragmentation equations. J. Math. Anal. App. 413 (2014) 1017–1029. | MR | DOI
[14] , and , Strong fragmentation and coagulation with power-law rates. J. Eng. Math. 82 (2013) 199–215. | MR | DOI
[15] , and , Analysis and simulations of the discrete fragmentation equation with decay. Math. Methods Appl. Sci. 41 (2018) 6530–6545. | MR | DOI
[16] and , Improving the accuracy of the moments method for solving the aerosol general dynamic equation. J. Aerosol Sci. 27 (1996) 1135–1142. | DOI
[17] , and , Coagulation and fragmentation with discrete mass loss. J. Math. Anal. Appl. 329 (2007) 1285–1302. | MR | Zbl | DOI
[18] and , Convergence of a finite volume scheme for coagulation–fragmentation equations. Math. Comp. 77 (2008) 851–882. | MR | Zbl | DOI
[19] , Evolution equations in a Banach space (2006). http://canizo.org/tex/evolution.pdf.
[20] , and , Some analytical and numerical solutions for colloidal aggregation with fragmentation. J. Phys. A: Math. General 28 (1995) 2981–2994. | MR | DOI
[21] and , Numerical approximation of a coagulation–fragmentation model for animal group size statistics. Netw. Heterogeneous Media 12 (2017) 217–243. | MR | DOI
[22] , and , Coagulation–fragmentation model for animal group-size statistics. J. Nonlinear Sci. 27 (2017) 379–424. | MR | DOI
[23] , and , Modeling dust growth in protoplanetary disks: the breakthrough case. Astron. Astrophys. 567 (2014) A38. | DOI
[24] and , An efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena. SIAM J. Sci. Comput. 22 (2000) 802–821. | MR | Zbl | DOI
[25] and , One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000). | MR | Zbl
[26] , An asymptotically stable scheme for diffusive coagulation–fragmentation models. Commun. Math. Sci. 6 (2008) 257–280. | MR | Zbl | DOI
[27] and , Numerical simulation of the Smoluchowski coagulation equation. SIAM J. Sci. Comput. 25 (2004) 2004–2028. | MR | Zbl | DOI
[28] , On the distribution of the sizes of particles which undergo splitting. Theory Probab. App. 6 (1961) 275–294. | Zbl | DOI
[29] and , Modern Methods in the Calculus of Variations: Spaces. Springer, New York (2007). | MR | Zbl
[30] and , A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence. SIAM J. Sci. Comput. 34 (2012) B840–B860. | MR | Zbl | DOI
[31] , A Monte Carlo approach to the Smoluchowski equations. Monte Carlo Methods App. 3 (1997) 313–326. | MR | Zbl
[32] , , and , A coagulation–fragmentation model for the turbulent growth and destruction of preplanetesimals. Astron. Astrophys. 486 (2008) 597–611. | Zbl | DOI
[33] , and , Development and convergence analysis of a finite volume scheme for solving breakage equation. SIAM J. Numer. Anal. 53 (2015) 1672–1689. | MR | DOI
[34] , Numerical analysis of finite volume schemes for population balance equations. Ph.D. thesis, Otto-von-Guericke University Magdeburg (2010). https://www.researchgate.net/publication/50870176_Numerical_analysis_for_finite_volume_schemes_for_population_balance_equations.
[35] and , On the solution of population balance equations by discretization–I. A fixed pivot technique. Chem. Eng. Sci. 51 (1996) 1311–1332. | DOI
[36] , and , Moment preserving finite volume schemes for solving population balance equations incorporating aggregation, breakage, growth and source terms. Math. Models Methods Appl. Sci. 23 (2013) 1235–1273. | MR | Zbl | DOI
[37] , and , Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinet. Related Models 7 (2014) 713–737. | MR | Zbl | DOI
[38] , Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math. Meth. Appl. Sci. 27 (2004) 703–721. | MR | Zbl | DOI
[39] , Weak compactness techniques and coagulation equations. In: Vol. 2126 of Evolutionary Equations with Applications in Natural Sciences. Lecture Notes in Math. Springer International Publishing (2015) 199–253. | MR | DOI
[40] and , Analysis. Graduate Studies in Mathematics. American Mathematical Society (2001). | MR | Zbl | DOI
[41] , and , Strongly differentiable solutions of the discrete coagulation–fragmentation equation. Phys. D: Nonlinear Phenom. 239 (2010) 1436–1445. | MR | Zbl | DOI
[42] and , Shattering transition in fragmentation. Phys. Rev. Lett. 58 (1987) 892–895. | MR | DOI
[43] , and , An existence and uniqueness theorem for a coagulation and multiple-fragmentation equation. SIAM J. Math. Anal. 28 (1997) 1173–1190. | MR | Zbl | DOI
[44] , and , A semigroup approach to fragmentation models. SIAM J. Math. Anal. 28 (1997) 1158–1172. | MR | Zbl | DOI
[45] , Towards a matrix theory for unbounded operator matrices. Math. Z. 201 (1989) 57–68. | MR | Zbl | DOI
[46] and , A finite element analysis of the steady state population balance equation for particulate systems: aggregation and growth. Comput. Chem. Eng. 20 (1996) S261–S266. | DOI
[47] , Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983). | MR | Zbl | DOI
[48] and , Solving population balance equations for two-component aggregation by a finite volume scheme. Chem. Eng. Sci. 62 (2007) 679–693. | DOI
[49] , and , On the solution of population balances for nucleation, growth, aggregation and breakage processes. Chem. Eng. Sci. 64 (2009) 2088–2095. | DOI
[50] , , and , Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinet. Related Models 12 (2019) 79–103. | MR | Zbl | DOI
[51] , , , and , Fast and accurate finite-difference method solving multicomponent Smoluchowski coagulation equation with source and sink terms. Proc. Comput. Sci. 80 (2016) 2141–2146. | DOI
[52] , , and , Discrete fragmentation with mass loss. J. Evol. Equ. 12 (2012) 181–201. | MR | Zbl | DOI
[53] , A global existence theorem for the general coagulation–fragmentation equation with unbounded kernels. Math. Methods Appl. Sci. 11 (1989) 627–648. | MR | Zbl | DOI
[54] , On substochastic -semigroups and their generators. Trans. Theo. Stat. Phys. 16 (1987) 453–466. | MR | Zbl | DOI
[55] , Kinetics of polymerization. J. Stat. Phys. 23 (1980) 241–263. | MR | DOI
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