• Previous Article
    Boltzmann-type equations for multi-agent systems with label switching
  • KRM Home
  • This Issue
  • Next Article
    Towards a further understanding of the dynamics in the excitatory NNLIF neuron model: Blow-up and global existence
October  2021, 14(5): 847-866. doi: 10.3934/krm.2021026

A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving

1. 

Sorbonne University, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

2. 

University of Würzburg, Germany

3. 

Department of Mathematics and Institute of Natural Sciences, Jiao Tong Univ., Shanghai, China

* Corresponding author: Farah Kanbar

Received  July 2020 Revised  May 2021 Published  October 2021 Early access  August 2021

Fund Project: The second author is supported by the National Council for Scientific Research of Lebanon (CNRS-L). The second and third authors are supported by Bayerisches Hochschulzentrum for China

In this work we are interested in the stationary preserving property of asymptotic preserving (AP) schemes for kinetic models. We introduce a criterion for AP schemes for kinetic equations to be uniformly stationary preserving (SP). Our key observation is that as long as the Maxwellian of the distribution function can be updated explicitly, such AP schemes are also SP. To illustrate our observation, three different AP schemes for three different kinetic models are considered. Their SP property is proved analytically and tested numerically, which confirms our observations.

Citation: Casimir Emako, Farah Kanbar, Christian Klingenberg, Min Tang. A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving. Kinetic and Related Models, 2021, 14 (5) : 847-866. doi: 10.3934/krm.2021026
References:
[1]

M. Adams, Discontinuous finite element transport solutions in thick diffusive problems, Nuclear Science and Engineering, 137 (2001), 298-333.  doi: 10.13182/NSE00-41.

[2]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.

[3]

S. Boscarino and L. Pareschi, On the asymptotic properties of IMEX Runge–Kutta schemes for hyperbolic balance laws, J. Comput. Appl. Math., 316 (2017), 60-73.  doi: 10.1016/j.cam.2016.08.027.

[4]

R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math. Comp., 72 (2003), 131–157 (electronic). doi: 10.1090/S0025-5718-01-01371-0.

[5]

J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Model. Simul., 11 (2013), 336-361.  doi: 10.1137/110851687.

[6]

C. Cercignani, The Boltzmann equation and its applications, Springer, 1988. doi: 10.1007/978-1-4612-1039-9.

[7]

F. A. C. C. ChalubP. A. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.

[8]

A. ChertockA. KurganovM. Lukáčová-Medvid'ová and S.N. Özcan, An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions, Kinet. Relat. Models, 12 (2019), 195-216.  doi: 10.3934/krm.2019009.

[9]

G. Dimarco and L. Pareschi, Implicit-explicit linear multistep methods for stiff kinetic equations, SIAM J. Numer. Anal., 55 (2017), 664-690.  doi: 10.1137/16M1063824.

[10]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, Journal of Computational Physics, 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.

[11]

F. FilbetC. Mouhot and L. Pareschi, Solving the Boltzmann equation in $n\log_2n$, SIAM Journal of Scientific Computation, 28 (2006), 1029-1053. 

[12]

L. Gosse, A well-balanced scheme for kinetic models of chemotaxis derived from one-dimensional local forward-backward problems, Math. Biosci., 242 (2013), 117-128.  doi: 10.1016/j.mbs.2012.12.009.

[13]

L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337-342.  doi: 10.1016/S1631-073X(02)02257-4.

[14]

L. Gosse and G. Toscani, Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2003), 641-658.  doi: 10.1137/S0036142901399392.

[15]

L. Gosse and G. Toscani, Asymptotic-preserving & well-balanced schemes for radiative transfer and the rosseland approximation, Numer. Math., 98 (2004), 223-250.  doi: 10.1007/s00211-004-0533-x.

[16]

S. Guisset, S. Brull, E. D'Humières and B. Dubroca, Asymptotic-preserving well-balanced scheme for the electronic $M_1$ model in the diffusive limit: Particular cases, ESAIM Math. Model. Numer. Anal., 51 (2017), 1805-1826. doi: 10.1051/m2an/2016079.

[17]

B. Howard, E. Coli in Motion, Biological and Medical Physics, Biomedical Engineering, Springer, 2004.

[18]

J. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator, Commun. Math. Sci., 10 (2012), 989-999.  doi: 10.4310/CMS.2012.v10.n3.a13.

[19]

H. J. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: a generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.

[20]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Math. Univ. Parma (N.S.), 3 (2012), 177-216. 

[21]

S. JinL. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), 913-936.  doi: 10.1137/S0036142998347978.

[22]

S. JinM. Tang and H. Han, A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface, Netw. Heterog. Media, 4 (2009), 35-65.  doi: 10.3934/nhm.2009.4.35.

[23]

E. W. Larsen and J. E. Morel, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II, J. Comput. Phys., 83 (1989), 212-236.  doi: 10.1016/0021-9991(89)90229-5.

[24]

E. W. LarsenJ. E. Morel and W. F. Jr Miller, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes, J. Comput. Phys., 69 (1987), 283-324.  doi: 10.1016/0021-9991(87)90170-7.

[25]

L. Mieussens, On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models, J. Comput. Phys., 253 (2013), 138-156.  doi: 10.1016/j.jcp.2013.07.002.

[26]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.

[27]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.

[28]

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12 pp. doi: 10.1371/journal. pcbi. 1000890.

[29]

J. SaragostiV. CalvezN. BournaveasB. PerthameA. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, Proceedings of the National Academy of Sciences, 108 (2011), 16235-16240.  doi: 10.1073/pnas.1101996108.

[30]

K. Xu, A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), 289-335.  doi: 10.1006/jcph.2001.6790.

[31]

K. Xu and J.-C. Huang, A unified gas-kinetic scheme for continuum and rarefied flows, J. Comput. Phys., 229 (2010), 7747-7764.  doi: 10.1016/j.jcp.2010.06.032.

show all references

References:
[1]

M. Adams, Discontinuous finite element transport solutions in thick diffusive problems, Nuclear Science and Engineering, 137 (2001), 298-333.  doi: 10.13182/NSE00-41.

[2]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.

[3]

S. Boscarino and L. Pareschi, On the asymptotic properties of IMEX Runge–Kutta schemes for hyperbolic balance laws, J. Comput. Appl. Math., 316 (2017), 60-73.  doi: 10.1016/j.cam.2016.08.027.

[4]

R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math. Comp., 72 (2003), 131–157 (electronic). doi: 10.1090/S0025-5718-01-01371-0.

[5]

J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Model. Simul., 11 (2013), 336-361.  doi: 10.1137/110851687.

[6]

C. Cercignani, The Boltzmann equation and its applications, Springer, 1988. doi: 10.1007/978-1-4612-1039-9.

[7]

F. A. C. C. ChalubP. A. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.

[8]

A. ChertockA. KurganovM. Lukáčová-Medvid'ová and S.N. Özcan, An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions, Kinet. Relat. Models, 12 (2019), 195-216.  doi: 10.3934/krm.2019009.

[9]

G. Dimarco and L. Pareschi, Implicit-explicit linear multistep methods for stiff kinetic equations, SIAM J. Numer. Anal., 55 (2017), 664-690.  doi: 10.1137/16M1063824.

[10]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, Journal of Computational Physics, 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.

[11]

F. FilbetC. Mouhot and L. Pareschi, Solving the Boltzmann equation in $n\log_2n$, SIAM Journal of Scientific Computation, 28 (2006), 1029-1053. 

[12]

L. Gosse, A well-balanced scheme for kinetic models of chemotaxis derived from one-dimensional local forward-backward problems, Math. Biosci., 242 (2013), 117-128.  doi: 10.1016/j.mbs.2012.12.009.

[13]

L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337-342.  doi: 10.1016/S1631-073X(02)02257-4.

[14]

L. Gosse and G. Toscani, Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2003), 641-658.  doi: 10.1137/S0036142901399392.

[15]

L. Gosse and G. Toscani, Asymptotic-preserving & well-balanced schemes for radiative transfer and the rosseland approximation, Numer. Math., 98 (2004), 223-250.  doi: 10.1007/s00211-004-0533-x.

[16]

S. Guisset, S. Brull, E. D'Humières and B. Dubroca, Asymptotic-preserving well-balanced scheme for the electronic $M_1$ model in the diffusive limit: Particular cases, ESAIM Math. Model. Numer. Anal., 51 (2017), 1805-1826. doi: 10.1051/m2an/2016079.

[17]

B. Howard, E. Coli in Motion, Biological and Medical Physics, Biomedical Engineering, Springer, 2004.

[18]

J. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator, Commun. Math. Sci., 10 (2012), 989-999.  doi: 10.4310/CMS.2012.v10.n3.a13.

[19]

H. J. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: a generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.

[20]

S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Math. Univ. Parma (N.S.), 3 (2012), 177-216. 

[21]

S. JinL. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), 913-936.  doi: 10.1137/S0036142998347978.

[22]

S. JinM. Tang and H. Han, A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface, Netw. Heterog. Media, 4 (2009), 35-65.  doi: 10.3934/nhm.2009.4.35.

[23]

E. W. Larsen and J. E. Morel, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II, J. Comput. Phys., 83 (1989), 212-236.  doi: 10.1016/0021-9991(89)90229-5.

[24]

E. W. LarsenJ. E. Morel and W. F. Jr Miller, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes, J. Comput. Phys., 69 (1987), 283-324.  doi: 10.1016/0021-9991(87)90170-7.

[25]

L. Mieussens, On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models, J. Comput. Phys., 253 (2013), 138-156.  doi: 10.1016/j.jcp.2013.07.002.

[26]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.

[27]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250.  doi: 10.1137/S0036139900382772.

[28]

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., 6 (2010), e1000890, 12 pp. doi: 10.1371/journal. pcbi. 1000890.

[29]

J. SaragostiV. CalvezN. BournaveasB. PerthameA. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, Proceedings of the National Academy of Sciences, 108 (2011), 16235-16240.  doi: 10.1073/pnas.1101996108.

[30]

K. Xu, A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), 289-335.  doi: 10.1006/jcph.2001.6790.

[31]

K. Xu and J.-C. Huang, A unified gas-kinetic scheme for continuum and rarefied flows, J. Comput. Phys., 229 (2010), 7747-7764.  doi: 10.1016/j.jcp.2010.06.032.

Figure 1.  Neutron Transport. Left: The density $ \rho_{r} $ at time $ t = 0.05 $ for $ \varepsilon = 10^{-2} $, $ \varepsilon = 10^{-3} $, $ \varepsilon = 10^{-6} $ and the solution of the diffusion limit equation; right: a zoomed part of the left plot
Figure 2.  Neutron Transport: Time evolution of the $ L^\infty $-norm of the difference between the solution and the stationary solution in the time interval [0, 8] for $ \varepsilon = 10^{-8} $
Figure 3.  Verification of the AP property of the UGKS for the chemotaxis kinetic model. Left: The density $ \rho $ at time $ t = 1 $ for $ \varepsilon = 10^{-2},10^{-3},10^{-4},10^{-5},10^{-6} $; right: a zoomed part of the left plot
Figure 4.  Boltzmann: cross section of the distribution function for different values of $ \varepsilon $(left) and a zoomed part of the plot(right)
Figure 5.  Boltzmann: contours of the 2D distribution function (left) and the contour lines of the difference between the distribution function and its equilibrium (right) at the final time $ t = 200 $
Figure 6.  Boltzmann: Time evolution of the $ L^\infty $-norm of the difference between the distribution function $ f $ and its equilibrium in the time interval [0, 200]
Table 1.  A list of kinetic models together with their corresponding schemes
Section Kinetic Model Scheme Reference
2 Neutron transport equation Parity-equations based [20]
3 Chemotaxis kinetic model UGKS [25, 30, 31]
4 Boltzmann equation IMEX Penalization method [10]
Section Kinetic Model Scheme Reference
2 Neutron transport equation Parity-equations based [20]
3 Chemotaxis kinetic model UGKS [25, 30, 31]
4 Boltzmann equation IMEX Penalization method [10]
Table 2.  Neutron Transport: $ L^\infty $-norm of the difference between the solution and the stationary solution in the time interval [0, 8] for $ \varepsilon = 10^{-2} $ (top) and $ \varepsilon = 10^{-8} $ (bottom)
T 0 2 4 6 8
$ L^\infty $ 0.995 $ 1.051\times 10^{-3} $ $ 1.683\times 10^{-6} $ $ 2.696\times 10^{-9} $ $ 4.120\times 10^{-12} $
T 0 2 4 6 8
$ L^\infty $ 1 $ 9.111\times 10^{-4} $ $ 1.263\times 10^{-6} $ $ 1.752\times 10^{-9} $ $ 2.176\times 10^{-12} $
T 0 2 4 6 8
$ L^\infty $ 0.995 $ 1.051\times 10^{-3} $ $ 1.683\times 10^{-6} $ $ 2.696\times 10^{-9} $ $ 4.120\times 10^{-12} $
T 0 2 4 6 8
$ L^\infty $ 1 $ 9.111\times 10^{-4} $ $ 1.263\times 10^{-6} $ $ 1.752\times 10^{-9} $ $ 2.176\times 10^{-12} $
Table 3.  Chemotaxis: $ L^\infty $-norm of the difference between the solution and the stationary solution in the time interval [0, 100] for $ \varepsilon = 1 $ (top) and $ \varepsilon = 10^{-3} $ (bottom)
T 0 30 60 65 100
$ L^\infty $ 0.9064 $ 8.260\times 10^{-7} $ $ 3.767\times 10^{-11} $ $ 7.474\times 10^{-12} $ $ 1.662\times 10^{-12} $
T 0 5 10 50 100
$ L^\infty $ 0.6493 $ 3.024\times 10^{-7} $ $ 2.064\times 10^{-9} $ $ 2.199\times 10^{-10} $ $ 1.476\times 10^{-10} $
T 0 30 60 65 100
$ L^\infty $ 0.9064 $ 8.260\times 10^{-7} $ $ 3.767\times 10^{-11} $ $ 7.474\times 10^{-12} $ $ 1.662\times 10^{-12} $
T 0 5 10 50 100
$ L^\infty $ 0.6493 $ 3.024\times 10^{-7} $ $ 2.064\times 10^{-9} $ $ 2.199\times 10^{-10} $ $ 1.476\times 10^{-10} $
Table 4.  Boltzmann: $ L^\infty $-norm of the difference between $ f $ and its equilibrium starting from t = 0 till the final time t = 200 for $ \varepsilon = 1 $
T 0 20 50 100 150 200
$ L^\infty $ 0.5453 $ 1.2\times 10^{-3} $ $ 6.581\times 10^{-7} $ $ 3.495\times 10^{-12} $ 7.619 $ \times 10^{-13} $ 5.623$ \times 10^{-13} $
T 0 20 50 100 150 200
$ L^\infty $ 0.5453 $ 1.2\times 10^{-3} $ $ 6.581\times 10^{-7} $ $ 3.495\times 10^{-12} $ 7.619 $ \times 10^{-13} $ 5.623$ \times 10^{-13} $
[1]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic and Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[2]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic and Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707

[3]

Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic and Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51

[4]

Alina Chertock, Changhui Tan, Bokai Yan. An asymptotic preserving scheme for kinetic models with singular limit. Kinetic and Related Models, 2018, 11 (4) : 735-756. doi: 10.3934/krm.2018030

[5]

Nicolas Crouseilles, Giacomo Dimarco, Mohammed Lemou. Asymptotic preserving and time diminishing schemes for rarefied gas dynamic. Kinetic and Related Models, 2017, 10 (3) : 643-668. doi: 10.3934/krm.2017026

[6]

Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146

[7]

Roman Romanov. Estimates of solutions of linear neutron transport equation at large time and spectral singularities. Kinetic and Related Models, 2012, 5 (1) : 113-128. doi: 10.3934/krm.2012.5.113

[8]

Lei Wu. Diffusive limit with geometric correction of unsteady neutron transport equation. Kinetic and Related Models, 2017, 10 (4) : 1163-1203. doi: 10.3934/krm.2017045

[9]

Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093

[10]

Jiaxiang Cai, Juan Chen, Min Chen. Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2441-2453. doi: 10.3934/dcdsb.2021139

[11]

Frederic Heihoff. Global mass-preserving solutions for a two-dimensional chemotaxis system with rotational flux components coupled with a full Navier–Stokes equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4703-4719. doi: 10.3934/dcdsb.2020120

[12]

Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789

[13]

Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic and Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955

[14]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic and Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139

[15]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[16]

Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206

[17]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147

[18]

Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.

[19]

Francesca R. Guarguaglini. Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks and Heterogeneous Media, 2018, 13 (1) : 47-67. doi: 10.3934/nhm.2018003

[20]

Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic and Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (131)
  • HTML views (187)
  • Cited by (0)

[Back to Top]