American Institute of Mathematical Sciences

Advanced
March  2006, 1(1): 143-166. doi: 10.3934/nhm.2006.1.143

Numerical study of a domain decomposition method for a two-scale linear transport equation

Xu Yang 1, , François Golse 2, , Zhongyi Huang 3, and  Shi Jin 1,

1. 

Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States, United States
2. 

Institut Universitaire de France & Département de Mathématiques et Applications, Ecole Normale Supérieure Paris, 45 rue d'Ulm, 75230 Paris cedex 05, France
3. 

Dept. of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received  September 2005 Revised  November 2005 Published  January 2006

We perform a numerical study on a domain decomposition method proposed in [13] for the linear transport equation between a diffusive and a non-diffusive region. This method avoids iterating the diffusion and transport solutions as in a typical domain decomposition method. Our numerical results, in both one and two space dimensions, confirm the theoretical analysis of [13]. We also provide an improved second order method that provides a more accurate numerical solution than that proposed in [13].
Keywords: domain decomposition, Linear transport equation, numerical approximation., diffusion approximation.
Mathematics Subject Classification: 35B25, 65M60, 76R50, 82C7.
Citation: Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks & Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143
[1]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[2]

Naoufel Ben Abdallah, Raymond El Hajj. Diffusion and guiding center approximation for particle transport in strong magnetic fields. Kinetic & Related Models, 2008, 1 (3) : 331-354. doi: 10.3934/krm.2008.1.331
[3]

Caojin Zhang, George Yin, Qing Zhang, Le Yi Wang. Pollution control for switching diffusion models: Approximation methods and numerical results. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3667-3687. doi: 10.3934/dcdsb.2018310
[4]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65
[5]

Yves Achdou, Fabio Camilli, Lucilla Corrias. On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 629-650. doi: 10.3934/dcdsb.2014.19.629
[6]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
[7]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[8]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163
[9]

Antonio DeSimone, Martin Kružík. Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation. Networks & Heterogeneous Media, 2013, 8 (2) : 481-499. doi: 10.3934/nhm.2013.8.481
[10]

Pedro Aceves-Sánchez, Christian Schmeiser. Fractional diffusion limit of a linear kinetic equation in a bounded domain. Kinetic & Related Models, 2017, 10 (3) : 541-551. doi: 10.3934/krm.2017021
[11]

Jakub Cupera. Diffusion approximation of neuronal models revisited. Mathematical Biosciences & Engineering, 2014, 11 (1) : 11-25. doi: 10.3934/mbe.2014.11.11
[12]

Nicolas Lecoq, Helena Zapolsky, P.K. Galenko. Numerical approximation of the Chan-Hillard equation with memory effects in the dynamics of phase separation. Conference Publications, 2011, 2011 (Special) : 953-962. doi: 10.3934/proc.2011.2011.953
[13]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497
[14]

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133
[15]

Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003
[16]

Fethallah Benmansour, Guillaume Carlier, Gabriel Peyré, Filippo Santambrogio. Numerical approximation of continuous traffic congestion equilibria. Networks & Heterogeneous Media, 2009, 4 (3) : 605-623. doi: 10.3934/nhm.2009.4.605
[17]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[18]

Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure & Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018
[19]

T. Hillen. On the $L^2$-moment closure of transport equations: The Cattaneo approximation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 961-982. doi: 10.3934/dcdsb.2004.4.961
[20]

Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences