• Previous Article
    System response of an alcoholism model under the effect of immigration via non-singular kernel derivative
  • DCDS-S Home
  • This Issue
  • Next Article
    Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion
July  2021, 14(7): 2183-2197. doi: 10.3934/dcdss.2021062

A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics

MAE2D laboratory, University Abdelmalek Essaadi, Polydisciplinary Faculty of Larache, Road of Rabat, Larache, Morocco

* Corresponding author: anouarelharrak1@gmail.com

Received  December 2019 Revised  June 2020 Published  July 2021 Early access  May 2021

This work gives a posteriori error estimates for a finite volume implicit scheme, applied to a two-time nonlinear reaction-diffusion problem in population dynamics, whose evolution processes occur at two different time scales, represented by a parameter $ \varepsilon>0 $ small enough. This work consists of building error indicators concerning time and space approximations and using them as a tool of adaptive mesh refinement in order to find approximate solutions to such models, in population dynamics, that are often hard to be handled analytically and also to be approximated numerically using the classical approach.

An application of the theoretical results is provided to emphasize the efficiency of our approach compared to the classical one for a spatial inter-specific model with constant diffusivity and population growth given by a logistic law in population dynamics.

Citation: Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062
References:
[1]

B. AmazianeA. BergamM. El Ossmani and Z. Mghazli, A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media, J. Numer. Meth. Fluids, 59 (2009), 259-284.  doi: 10.1002/fld.1456.

[2]

P. Auger and R. Bravo, Methods of aggregation of variables in population dynamics, Comptes Rendus de l'Académie des Sciences - Series Ⅲ - Sciences de la Vie, 323 (2000), 665-674.  doi: 10.1016/S0764-4469(00)00182-7.

[3]

P. AugerJ. C. Poggiale and E. Sánchez, A review on spatial aggregation methods involving several time scales, Ecological Complexity, 10 (2012), 12-25.  doi: 10.1016/j.ecocom.2011.09.001.

[4]

A. BergamC. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Math. Comp, 74 (2005), 1117-1138.  doi: 10.1090/S0025-5718-04-01697-7.

[5]

A. BergamA. ChakibA. Nachaoui and M. Nachaoui, Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem, Applied Mathematics and Computation, 346 (2019), 865-878.  doi: 10.1016/j.amc.2018.09.069.

[6]

A. El Harrak and A. Bergam, Preserving Finite-Volume Schemes for Two-Time Reaction-Diffusion Model, Applied Mathematics & Information Sciences, 14 (2020), 41-50.  doi: 10.18576/amis/140105.

[7]

R. EymardT. Gallouöt and R. Herbin, Finite volume methods, Hand-book of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds, North-Holland, (2000), 713-1020.  doi: 10.1086/phos.67.4.188705.

[8]

M. Fortin, Estimation a posteriori et adaptation de maillages, Revue Européenne des Eléments Finis, 9 (2000).

[9] W. Gurney and R. M. Nisbet, Ecological Dynamics,, Oxford University Press, 1998. 
[10]

Z. MghazliR. Verfürth and A. Bergam, Estimateurs a posteriori d'un schéma de volumes finis pour un probléme non linéaire, Numerische Mathematik, 95 (2003), 599-624.  doi: 10.1007/s00211-003-0460-2.

[11]

J. D. Murray, Mathematical Biology: Ⅰ. an Introduction (Interdisciplinary Applied Mathematics), , Springer-Verlag, New York, 2002.

[12]

E. SánchezP. Auger and J. C. Poggiale, Two-time scales in spatially structured models of population dynamics: A semigroup approach, Journal of Mathematical Analysis and Applications, 375 (2011), 149-165.  doi: 10.1016/j.jmaa.2010.08.014.

[13]

D. Tilman and P. Kareiva, Spatial ecology: The role of space in population dynamics and interspecific interactions (MPB-30), Princeton University Press, 30 (2018).

[14]

R. Verfürth, A posteriori error estimates for non-stationary non-linear convection-diffusion equations, Calcolo, 55 (2018), Paper No. 20, 18 pp. doi: 10.1007/s10092-018-0263-6.

show all references

References:
[1]

B. AmazianeA. BergamM. El Ossmani and Z. Mghazli, A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media, J. Numer. Meth. Fluids, 59 (2009), 259-284.  doi: 10.1002/fld.1456.

[2]

P. Auger and R. Bravo, Methods of aggregation of variables in population dynamics, Comptes Rendus de l'Académie des Sciences - Series Ⅲ - Sciences de la Vie, 323 (2000), 665-674.  doi: 10.1016/S0764-4469(00)00182-7.

[3]

P. AugerJ. C. Poggiale and E. Sánchez, A review on spatial aggregation methods involving several time scales, Ecological Complexity, 10 (2012), 12-25.  doi: 10.1016/j.ecocom.2011.09.001.

[4]

A. BergamC. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Math. Comp, 74 (2005), 1117-1138.  doi: 10.1090/S0025-5718-04-01697-7.

[5]

A. BergamA. ChakibA. Nachaoui and M. Nachaoui, Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem, Applied Mathematics and Computation, 346 (2019), 865-878.  doi: 10.1016/j.amc.2018.09.069.

[6]

A. El Harrak and A. Bergam, Preserving Finite-Volume Schemes for Two-Time Reaction-Diffusion Model, Applied Mathematics & Information Sciences, 14 (2020), 41-50.  doi: 10.18576/amis/140105.

[7]

R. EymardT. Gallouöt and R. Herbin, Finite volume methods, Hand-book of Numerical Analysis, P.G. Ciarlet and J.L. Lions eds, North-Holland, (2000), 713-1020.  doi: 10.1086/phos.67.4.188705.

[8]

M. Fortin, Estimation a posteriori et adaptation de maillages, Revue Européenne des Eléments Finis, 9 (2000).

[9] W. Gurney and R. M. Nisbet, Ecological Dynamics,, Oxford University Press, 1998. 
[10]

Z. MghazliR. Verfürth and A. Bergam, Estimateurs a posteriori d'un schéma de volumes finis pour un probléme non linéaire, Numerische Mathematik, 95 (2003), 599-624.  doi: 10.1007/s00211-003-0460-2.

[11]

J. D. Murray, Mathematical Biology: Ⅰ. an Introduction (Interdisciplinary Applied Mathematics), , Springer-Verlag, New York, 2002.

[12]

E. SánchezP. Auger and J. C. Poggiale, Two-time scales in spatially structured models of population dynamics: A semigroup approach, Journal of Mathematical Analysis and Applications, 375 (2011), 149-165.  doi: 10.1016/j.jmaa.2010.08.014.

[13]

D. Tilman and P. Kareiva, Spatial ecology: The role of space in population dynamics and interspecific interactions (MPB-30), Princeton University Press, 30 (2018).

[14]

R. Verfürth, A posteriori error estimates for non-stationary non-linear convection-diffusion equations, Calcolo, 55 (2018), Paper No. 20, 18 pp. doi: 10.1007/s10092-018-0263-6.

Figure 1.  Plot of approximate solution of Problem (12) using uniform mesh, ($ A $) and ($ C $), and adaptive mesh, ($ B $) and ($ D $), at two instants $ t = 1 $ and $ t = 10 $
Figure 2.  Self-adapted meshes of three levels at instant $ t = 1 $
Figure 3.  Self-adapted meshes of three levels at instant $ t = 10 $
Table 1.  Numerical tests for Problem (12)
Instant Mesh Level Number of triangles CPU time Mean of $ \eta_h^n $
t=1 Adaptive 1 $ 256 $ $ 1.957s $ $ 3.1229e-2 $
2 $ 498 $ $ 2.321s $ $ 1.2628e-2 $
3 $ 1071 $ $ 5.492s $ $ 4.5722e-3 $
Uniform 1 $ 6145 $ $ 89.947s $ $ 3.3747e-3 $
t=10 Adaptive 1 $ 207 $ $ 9.861s $ $ 13053e-2 $
2 $ 504 $ $ 16.087s $ $ 7.8965e-3 $
3 $ 726 $ $ 23.411s $ $ 3.5019e-3 $
Uniform 1 $ 6145 $ $ 719.854s $ $ 1.1385e-3 $
t=50 Adaptive 1 $ 356 $ $ 44.871s $ $ 1.1824e-2 $
2 $ 603 $ $ 76.644s $ $ 7.5797e-3 $
3 $ 879 $ $ 120.120s $ $ 2.2937e-3 $
Uniform 1 $ 6145 $ $ 2686.813s $ $ 1.1418e-3 $
Instant Mesh Level Number of triangles CPU time Mean of $ \eta_h^n $
t=1 Adaptive 1 $ 256 $ $ 1.957s $ $ 3.1229e-2 $
2 $ 498 $ $ 2.321s $ $ 1.2628e-2 $
3 $ 1071 $ $ 5.492s $ $ 4.5722e-3 $
Uniform 1 $ 6145 $ $ 89.947s $ $ 3.3747e-3 $
t=10 Adaptive 1 $ 207 $ $ 9.861s $ $ 13053e-2 $
2 $ 504 $ $ 16.087s $ $ 7.8965e-3 $
3 $ 726 $ $ 23.411s $ $ 3.5019e-3 $
Uniform 1 $ 6145 $ $ 719.854s $ $ 1.1385e-3 $
t=50 Adaptive 1 $ 356 $ $ 44.871s $ $ 1.1824e-2 $
2 $ 603 $ $ 76.644s $ $ 7.5797e-3 $
3 $ 879 $ $ 120.120s $ $ 2.2937e-3 $
Uniform 1 $ 6145 $ $ 2686.813s $ $ 1.1418e-3 $
[1]

Hatim Tayeq, Amal Bergam, Anouar El Harrak, Kenza Khomsi. Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2557-2570. doi: 10.3934/dcdss.2020400

[2]

Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659

[3]

Zhao-Han Sheng, Tingwen Huang, Jian-Guo Du, Qiang Mei, Hui Huang. Study on self-adaptive proportional control method for a class of output models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 459-477. doi: 10.3934/dcdsb.2009.11.459

[4]

Michele V. Bartuccelli, K. B. Blyuss, Y. N. Kyrychko. Length scales and positivity of solutions of a class of reaction-diffusion equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 25-40. doi: 10.3934/cpaa.2004.3.25

[5]

Ya-Zheng Dang, Zhong-Hui Xue, Yan Gao, Jun-Xiang Li. Fast self-adaptive regularization iterative algorithm for solving split feasibility problem. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1555-1569. doi: 10.3934/jimo.2019017

[6]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks and Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401

[7]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[8]

Mostafa Bendahmane, Mauricio Sepúlveda. Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 823-853. doi: 10.3934/dcdsb.2009.11.823

[9]

Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial and Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255

[10]

Yuan Shen, Chang Liu, Yannian Zuo, Xingying Zhang. A modified self-adaptive dual ascent method with relaxed stepsize condition for linearly constrained quadratic convex optimization. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022101

[11]

Lili Ju, Wensong Wu, Weidong Zhao. Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669

[12]

M. González, J. Jansson, S. Korotov. A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems. Conference Publications, 2015, 2015 (special) : 525-532. doi: 10.3934/proc.2015.0525

[13]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001

[14]

Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526

[15]

Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817

[16]

M. Syed Ali, L. Palanisamy, Nallappan Gunasekaran, Ahmed Alsaedi, Bashir Ahmad. Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1465-1477. doi: 10.3934/dcdss.2020395

[17]

Michel Pierre, Didier Schmitt. Examples of finite time blow up in mass dissipative reaction-diffusion systems with superquadratic growth. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022039

[18]

Costică Moroşanu. Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1567-1587. doi: 10.3934/dcdss.2020089

[19]

Walter Allegretto, Yanping Lin, Ningning Yan. A posteriori error analysis for FEM of American options. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 957-978. doi: 10.3934/dcdsb.2006.6.957

[20]

Messoud Efendiev, Alain Miranville. Finite dimensional attractors for reaction-diffusion equations in $R^n$ with a strong nonlinearity. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 399-424. doi: 10.3934/dcds.1999.5.399

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (280)
  • HTML views (226)
  • Cited by (0)

Other articles
by authors

[Back to Top]