American Institute of Mathematical Sciences

Advanced
June 2017, 12(2): 217-243. doi: 10.3934/nhm.2017009

Numerical approximation of a coagulation-fragmentation model for animal group size statistics

Pierre Degond , and  Maximilian Engel

Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK

* Corresponding author: Pierre Degond: pdegond@imperial.ac.uk

Received  April 2016 Revised  January 2017 Published  May 2017

Fund Project: This paper entitled "Numerical approximation of a coagulation-fragmentation model for animal group size statistics" is licensed under a Creative Commons Attribution 3.0 Unported License. See http://creativecommons.org/licenses/by/3.0/.

We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equilibrium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al. [12], a Newton method and the resolution of the time-dependent problem. All three schemes are validated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates.

Keywords: Coagulation-fragmentation equation, numerics, convergence to equilibrium, fish schools, Newton method, Euler scheme.
Mathematics Subject Classification: Primary: 45J05, 49M15, 82B99, 92D50.
Citation: Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks & Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009
References:
[1]

E. Bonabeau and L. Dagorn, Possible universality in the size distribution of fish schools, Phys. Rev. E, 51 (1995), 5220-5223. doi: 10.1103/PhysRevE.51.R5220.

[2]

E. Bonabeau, L. Dagorn and P. Freon, Scaling in animal group-size distributions, Proc. Natl. Acad. Sci. USA, 96 (1999), 4472-4477. doi: 10.1073/pnas.96.8.4472.

[3]

J. P. Bourgade and F. Filbet, Convergence of a finite volume scheme for coagulation-fragmentation equations, Comm. Math. Sciences, 77 (2008), 851-882. doi: 10.4310/CMS.2008.v6.n2.a1.

[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.
[5]

P. Degond, J. G. Liu and R. L. Pego, Coagulation-fragmentation model for animal group-size statistics, J. Nonlinear Sci., 27 (2017), 379-424. doi: 10.1007/s00332-016-9336-3.

[6]

F. Filbet and P. Laurencot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132.

[7]

L. Forestier-Coste and S. Mancini, A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM J. Sci. Comput., 34 (2012), B840-B860. doi: 10.1137/110847998.

[8]

S. Gueron, The steady-state distributions of coagulation-fragmentation processes, J. Math. Biol., 37 (1998), 1-27. doi: 10.1007/s002850050117.

[9]

S. Gueron and S. A. Levin, The dynamics of group formations, Math. Biosc., 128 (1995), 243-264. doi: 10.1016/0025-5564(94)00074-A.

[10]

J. Kumar, G. Kaur and E. Tsotsas, An accurate and efficient discrete formulation of aggregation population balance equation, Kinet. Relat. Models, 9 (2016), 373-391. doi: 10.3934/krm.2016.9.373.

[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 Methods Appl. Sci., 23 (2013), 1235-1273. doi: 10.1142/S0218202513500085.

[12]

Q. Ma, A. Johansson and D. J. T. Sumpter, A first principles derivation of animal group size distributions, J. Theoret. Biol., 283 (2011), 35-43. doi: 10.1016/j.jtbi.2011.04.031.

[13]

A. W. Mahoney and D. Ramkrishna, Efficient solution of population balance equations with discontinuities by finite elements, Chem. Eng. Sci., 57 (2002), 1107-1119. doi: 10.1016/S0009-2509(01)00427-4.

[14]

M. Nicmanis and M. J. Hounslow, A finite element analysis of the steady state population balance equation for particulate systems: Aggregation and growth, Comput. Chem. Eng., 20 (1996), 261-266. doi: 10.1016/0098-1354(96)00054-3.

[15]

H. Niwa, Mathematical model for the size distributions of fish schools, Comp. Math. Appl., 32 (1996), 79-88. doi: 10.1016/S0898-1221(96)00199-X.

[16]

H. Niwa, School size statistics of fish, J. Theoret. Biol., 195 (1998), 351-361. doi: 10.1006/jtbi.1998.0801.

[17]

H. Niwa, Power-Law versus exponential distributions of animal group sizes, J. Theoret. Biol., 224 (2003), 451-457. doi: 10.1016/S0022-5193(03)00192-9.

[18]

H. Niwa, Space-irrelevant scaling law for fish school sizes, J. Theoret. Biol., 228 (2004), 347-357. doi: 10.1016/j.jtbi.2004.01.011.

[19]

A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, rocks, and herds, Adv. Biophys., 22 (1986), 1-94.

[20]

S. Rigopoulos and A. G. Jones, Finite-element scheme for solution of the dynamic population balance equation, AIChE Journal, 49 (2003), 1127-1139. doi: 10.1002/aic.690490507.

[21]

D. Verkoeijen, G. A. Pouw, G. M. H. Meesters and B. Scarlett, Population balances for particulate processes-a volume approach, Chem. Eng. Sci., 57 (2002), 2287-2303. doi: 10.1016/S0009-2509(02)00118-5.

show all references

References:
[1]

E. Bonabeau and L. Dagorn, Possible universality in the size distribution of fish schools, Phys. Rev. E, 51 (1995), 5220-5223. doi: 10.1103/PhysRevE.51.R5220.

[2]

E. Bonabeau, L. Dagorn and P. Freon, Scaling in animal group-size distributions, Proc. Natl. Acad. Sci. USA, 96 (1999), 4472-4477. doi: 10.1073/pnas.96.8.4472.

[3]

J. P. Bourgade and F. Filbet, Convergence of a finite volume scheme for coagulation-fragmentation equations, Comm. Math. Sciences, 77 (2008), 851-882. doi: 10.4310/CMS.2008.v6.n2.a1.

[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.
[5]

P. Degond, J. G. Liu and R. L. Pego, Coagulation-fragmentation model for animal group-size statistics, J. Nonlinear Sci., 27 (2017), 379-424. doi: 10.1007/s00332-016-9336-3.

[6]

F. Filbet and P. Laurencot, Numerical simulation of the Smoluchowski coagulation equation, SIAM J. Sci. Comput., 25 (2004), 2004-2028. doi: 10.1137/S1064827503429132.

[7]

L. Forestier-Coste and S. Mancini, A finite volume preserving scheme on nonuniform meshes and for multidimensional coalescence, SIAM J. Sci. Comput., 34 (2012), B840-B860. doi: 10.1137/110847998.

[8]

S. Gueron, The steady-state distributions of coagulation-fragmentation processes, J. Math. Biol., 37 (1998), 1-27. doi: 10.1007/s002850050117.

[9]

S. Gueron and S. A. Levin, The dynamics of group formations, Math. Biosc., 128 (1995), 243-264. doi: 10.1016/0025-5564(94)00074-A.

[10]

J. Kumar, G. Kaur and E. Tsotsas, An accurate and efficient discrete formulation of aggregation population balance equation, Kinet. Relat. Models, 9 (2016), 373-391. doi: 10.3934/krm.2016.9.373.

[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 Methods Appl. Sci., 23 (2013), 1235-1273. doi: 10.1142/S0218202513500085.

[12]

Q. Ma, A. Johansson and D. J. T. Sumpter, A first principles derivation of animal group size distributions, J. Theoret. Biol., 283 (2011), 35-43. doi: 10.1016/j.jtbi.2011.04.031.

[13]

A. W. Mahoney and D. Ramkrishna, Efficient solution of population balance equations with discontinuities by finite elements, Chem. Eng. Sci., 57 (2002), 1107-1119. doi: 10.1016/S0009-2509(01)00427-4.

[14]

M. Nicmanis and M. J. Hounslow, A finite element analysis of the steady state population balance equation for particulate systems: Aggregation and growth, Comput. Chem. Eng., 20 (1996), 261-266. doi: 10.1016/0098-1354(96)00054-3.

[15]

H. Niwa, Mathematical model for the size distributions of fish schools, Comp. Math. Appl., 32 (1996), 79-88. doi: 10.1016/S0898-1221(96)00199-X.

[16]

H. Niwa, School size statistics of fish, J. Theoret. Biol., 195 (1998), 351-361. doi: 10.1006/jtbi.1998.0801.

[17]

H. Niwa, Power-Law versus exponential distributions of animal group sizes, J. Theoret. Biol., 224 (2003), 451-457. doi: 10.1016/S0022-5193(03)00192-9.

[18]

H. Niwa, Space-irrelevant scaling law for fish school sizes, J. Theoret. Biol., 228 (2004), 347-357. doi: 10.1016/j.jtbi.2004.01.011.

[19]

A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, rocks, and herds, Adv. Biophys., 22 (1986), 1-94.

[20]

S. Rigopoulos and A. G. Jones, Finite-element scheme for solution of the dynamic population balance equation, AIChE Journal, 49 (2003), 1127-1139. doi: 10.1002/aic.690490507.

[21]

D. Verkoeijen, G. A. Pouw, G. M. H. Meesters and B. Scarlett, Population balances for particulate processes-a volume approach, Chem. Eng. Sci., 57 (2002), 2287-2303. doi: 10.1016/S0009-2509(02)00118-5.

Figure 1.  The equilibrium distribution is approximated by the Newton scheme (Section 5.2.3). In Fig. 1a, we take mass $m_1 =1$, grid size $h=0.01$ and the cut-off at $L=100$. The plot shows the generated distribution (blue solid line) in a log scale against the group sizes in a linear scale and presents the theoretically found large-size asymptotic behaviour (red dashed line) in a log scale for the sake of comparison. The group sizes are taken in a linear scale in order to illustrate the leading behaviour for large group sizes as a straight line. For Fig. 1b, the equilibrium distribution is approximated by the Newton scheme taking mass $m_1 =1$, grid size $h=0.0005$ and the cut-off at $L=5$. The plot shows the generated distribution (blue solid line) in a log scale and presents the theoretically found asymptotic small-size behaviour (red dashed line) in a log scale for the sake of comparison. The group sizes are taken in a log scale as well in order to illustrate the leading behaviour close to zero as a straight line
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The equilibrium distribution is approximated by simulating the time evolution of the distribution via the Euler scheme. Starting with a uniform distribution, the equilibrium, is reached at $T=30$ at the latest. In Fig. 2a, we take mass $m_1 =1$, grid size $h=0.01$ and the cut-off at $L=100$. The plot shows the generated distribution (blue solid line) in a log scale as a function of the group sizes in a linear scale and presents the theoretically found large-size asymptotic behaviour (red dashed line) in a log scale for the sake of comparison. The group sizes are taken in a linear scale in order to illustrate the leading behaviour for large group sizes as a straight line. For Fig. 2b, the equilibrium distribution is approximated by the Euler scheme taking mass $m_1 =1$, grid size $h=0.0005$ and the cut-off at $L=5$. The plot shows the generated distribution (blue solid line) in a log scale and presents the theoretically found asymptotic small-size behaviour (red dashed line) in a log scale for the sake of comparison. The group sizes are taken in a log scale as well in order to illustrate the leading behaviour close to zero as a straight line
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The equilibrium distribution is approximated by the recursive scheme (Section 5.1). In Fig. 3a, the mass is $m_1^h =1$, the grid size $h=1$ and the equilibrium sequence is computed till $L=100$. It shows the generated distribution (blue solid line) in a log scale and presents the theoretically found asymptotic behaviour (red dashed line) in a log scale (Eq. 6.3) for the sake of comparison. One can observe perfect agreement for large sizes. In Fig. 3b, exactly the same is done for grid size $h=0.01$. Again, one can observe that the generated distribution shows the predicted asymptotics
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Comparison of the equilibria for model D' ((5.5)-(5.6)) and model D ((2.15)-(2.18)). We take truncation $L=100$, grid size $h=0.01$ and mass $m_1 = m_1^h =1$. The equilibrium for model D' is generated by the Newton scheme (Section 5.2.3) and represented in a log scale by the solid blue line. The equilibrium for model D is generated by the recursive scheme (Section 5.1) and represented in a log scale by the dotted red line
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The equilibrium distribution is generated by the recursive scheme, for mass $m_1 =1$, taking grid size $h=1$, $h=0.1$ and $h=0.01$. The figure shows the generated distributions (solid lines) and the large-size asymptotic behaviour for model C ((2.19)-(2.23)) (dashed line) in a log scale (equation (6.2)). We have magnified the plot close to $x=200$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The large-size behaviours of the discrete and continuous equilibrium distributions are compared, for mass $m_1 =1$ and fixed grid size $h=0.01$, close to $x=200$ (Fig. 6a), close to $x=1000$(Fig. 6b) and close to $x=2000$ (Fig. 6c). In each case, it shows the large-size asymptotic behaviour for model D ((2.15)-(2.18)) given by equation (6.3) (blue dotted line) and the large-size asymptotic behaviour for model C ((2.19)-(2.23)) given by equation (6.2) (red dashed line) in a log scale. Observe that for $x$ becoming greater, the difference between both graphs increases significantly
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  In Fig. 7a we plot $\frac{1}{h} f_1^h$ for $h \in [5*10^{-5}, 1]$ in log-log scale (blue solid line) and the small-size asymptotics of the continuous model C ((2.19)-(2.23)) (red dashed line). For small $h$, the graphs illustrate the findings in (6.6). In Fig. 7b, the equilibrium sequence for model D ((2.15)-(2.18)) is generated as described in Section 5.1 taking mass $m_1^h =1$ and grid size $h = 5*10^{-5}$. The plot shows the distribution $(f_i^h)_{i \in \mathbb{N}} $ as a function of the group size in log-log scale (blue solid line) in the interval $[h,1]$ and the small-size asymptotics of the continuous model C (red dashed line). Both graphs tend to have the same slope for the sizes becoming smaller except for a slight divergence at the smallest group sizes
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Starting with a uniform distribution (Fig. 8a) and with an exponential distribution (Fig. 8b), the time-dependent solution of model C ((2.19)-(2.23)), $f(x,t)$, is approximated via the Euler scheme for model D' ((5.5)-(5.6)), taking $L=100$, $h=0.01$, $m_1 =1$ and $\Delta t=1$ for uniform initial and $\Delta t = 0.5$ for exponential initial (due to stability issues for small sizes). The approximation, $f_i(t)$, is evaluated at $t = 20,\dots, 29$ and the equilibrium distribution is approximated via following the Euler scheme until $t=30$. Calculating the relative distances to the equilibrium, $ \mu_i(t) = \left|f_i^{\infty} - f_i(t)\right|/f_i^{\infty} $, for $i = 500$ and $i=9500$ (representing $x=5$ and $x=95$), we estimate the exponential convergence rate $\delta_{x,t_2}$ ($\sim \delta_{x,t_1}$) for $t_1 = 20,\dots, 28$ and $t_2 = t_1 +1$ according to Eq. (6.8)
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Starting with a uniform distribution the time-dependent solution of model C ((2.19)-(2.23)), $f(x,t)$, is approximated via the Euler scheme for model D' ((5.5)-(5.6)), taking $L=100$, $h=0.01$, $\Delta t=1$ and $m_1 =1$. This approximation, $f_i(t)$, is evaluated at $t = 5,10,15,20$ and the equilibrium distribution is approximated via following the Euler scheme until $t=30$. The table shows the relative distances to the equilibrium, $ \frac{\left|f_i^{\infty} - f_i(t)\right|}{f_i^{\infty} } $ for $t = 5,10,15,20,25$ and $i = 500,3500,6500,9500$
Time $t$ $x=5$ $x=35$ $x=65$ $x=95$
$t=5$ $0.2772$ $9.2148$ $154.7531$ $2046.0000$
$t=10$ $0.0638$ $1.4009$ $10.8288$ $67.0145$
$t=15$ $0.0089$ $0.1976$ $1.0721$ $3.8651 $
$t=20$ $0.0012$ $0.0260$ $0.1300$ $0.3832$
$t=25$ $0.0001$ $0.0030$ $0.0149$ $0.0423$
Table Options

Download as excel

Time $t$ $x=5$ $x=35$ $x=65$ $x=95$
$t=5$ $0.2772$ $9.2148$ $154.7531$ $2046.0000$
$t=10$ $0.0638$ $1.4009$ $10.8288$ $67.0145$
$t=15$ $0.0089$ $0.1976$ $1.0721$ $3.8651 $
$t=20$ $0.0012$ $0.0260$ $0.1300$ $0.3832$
$t=25$ $0.0001$ $0.0030$ $0.0149$ $0.0423$
Table 2.  Starting with an exponential distribution the time-dependent solution of model C ((2.19)-(2.23)), $f(x,t)$, is approximated via the Euler scheme for model D' ((5.5)-(5.6)), taking $L=100$, $h=0.01$, $\Delta t=0.5$ (smaller than in the previous case due to stabilisation problems for small sizes) and $m_1 =1$. This approximation, $f_i(t)$, is evaluated at $t = 5,10,15,20$ and the equilibrium distribution is approximated via following the Euler scheme until $t=30$. The table shows the relative distances to the equilibrium, $ \frac{\left|f_i^{\infty} - f_i(t)\right|}{f_i^{\infty} } $ for $t = 5,10,15,20,25$ and $i = 500,3500,6500,9500$
Time $t$ $x=5$ $x=35$ $x=65$ $x=95$
$t=5$ $0.05620$ $0.75370$ $0.98890$ $0.99980$
$t=10$ $0.00800$ $0.16010$ $0.50000$ $0.77500$
$t=15$ $0.00120$ $0.02670$ $0.11420$ $0.25710 $
$t=20$ $0.00020$ $0.00430$ $0.02000$ $0.05220$
$t=25$ $0.00003$ $0.00004$ $0.00290$ $0.00790$
Table Options

Download as excel

Time $t$ $x=5$ $x=35$ $x=65$ $x=95$
$t=5$ $0.05620$ $0.75370$ $0.98890$ $0.99980$
$t=10$ $0.00800$ $0.16010$ $0.50000$ $0.77500$
$t=15$ $0.00120$ $0.02670$ $0.11420$ $0.25710 $
$t=20$ $0.00020$ $0.00430$ $0.02000$ $0.05220$
$t=25$ $0.00003$ $0.00004$ $0.00290$ $0.00790$
[1]

Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043
[2]

Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 685-702. doi: 10.3934/cpaa.2010.9.685
[3]

Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541
[4]

Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic & Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014
[5]

Ankik Kumar Giri. On the uniqueness for coagulation and multiple fragmentation equation. Kinetic & Related Models, 2013, 6 (3) : 589-599. doi: 10.3934/krm.2013.6.589
[6]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[7]

Jacek Banasiak. Transport processes with coagulation and strong fragmentation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 445-472. doi: 10.3934/dcdsb.2012.17.445
[8]

Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563
[9]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
[10]

Eric A. Carlen, Süleyman Ulusoy. Localization, smoothness, and convergence to equilibrium for a thin film equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4537-4553. doi: 10.3934/dcds.2014.34.4537
[11]

Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial & Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333
[12]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126
[13]

Wilson Lamb, Adam McBride, Louise Smith. Coagulation and fragmentation processes with evolving size and shape profiles: A semigroup approach. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5177-5187. doi: 10.3934/dcds.2013.33.5177
[14]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1
[15]

Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173
[16]

Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic & Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493
[17]

Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995
[18]

Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1
[19]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[20]

Eric Cancès, Claude Le Bris. Convergence to equilibrium of a multiscale model for suspensions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 449-470. doi: 10.3934/dcdsb.2006.6.449

2017 Impact Factor: 1.187

Tools

Metrics

  • PDF downloads (20)
  • HTML views (227)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2018 American Institute of Mathematical Sciences