# American Institute of Mathematical Sciences

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

 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].
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] Victor Ginting. An adjoint-based a posteriori analysis of numerical approximation of Richards equation. Electronic Research Archive, 2021, 29 (5) : 3405-3427. doi: 10.3934/era.2021045 [7] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [8] 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 [9] 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 [10] 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 [11] 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 [12] Jakub Cupera. Diffusion approximation of neuronal models revisited. Mathematical Biosciences & Engineering, 2014, 11 (1) : 11-25. doi: 10.3934/mbe.2014.11.11 [13] 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 [14] R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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

2020 Impact Factor: 1.213