A domain decomposition method for semilinear hyperbolic systems with twoscale relaxations
Authors:
Shi Jin, Jianguo Liu and Li Wang
Journal:
Math. Comp. 82 (2013), 749779
MSC (2010):
Primary 65M55, 35L50
Published electronically:
October 9, 2012
MathSciNet review:
3008837
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Abstract: We present a domain decomposition method on a semilinear hyperbolic system with multiple relaxation times. In the region where the relaxation time is small, an asymptotic equilibrium equation can be used for computational efficiency. An interface condition based on the sign of the characteristic speed at the interface is provided to couple the two systems in a domain decomposition setting. A rigorous analysis, based on the Laplace Transform, on the error estimate is presented for the linear case, which shows how the error of the domain decomposition method depends on the smaller relaxation time, and the boundary and interface layer effects. The given convergence rate is optimal. We present a numerical implementation of this domain decomposition method, and give some numerical results in order to study the performance of this method.
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 [1]
 G. Bal, Y. Maday, Coupling of transport and diffusion models in linear transport theory, Math. Model. Numer. Anal. 36, no. 1, 6986, 2002. MR 1916293 (2003e:82065)
 [2]
 S. Bianchini, Hyperbolic limit of the JinXin relaxation model, Comm. Pure Applied Math. 59, 688753, 2006. MR 2172805 (2008b:35167)
 [3]
 J.F. Bourgat, P. Le Tallec, B. Perthame, Y. Qiu, Coupling Boltzmann and Euler equations without overlapping, in domain decomposition methods in science and engineering (Como, 1992), Contemp. Math. , Amer. Math. Soc. Providence, RI, 377398, 1994. MR 1262639 (95d:76085)
 [4]
 A. Bressan, Hyperbolic Systems of Conservation Laws: The OneDimensional Cauchy Problem, Oxford University Press, 2003. MR 1816648 (2002d:35002)
 [5]
 C. Cercignani, The Boltzmann Equation and Its Applications, SpringerVerlag, New York, 1988. MR 1313028 (95i:82082)
 [6]
 A. Chalabi, D. Seghir, Convergence of relaxation schemes for initial boundary value problems for conservation laws, Computers and Mathematics with Applications 43, no. 89, 10791093, 2002. MR 1892486 (2003a:35124)
 [7]
 G.Q. Chen, C.D. Levermore and T.P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47, 787830, 1994. MR 1280989 (95h:35133)
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 P. Degond, S. Jin, A smooth transition model between kinetic and diffusion equations, SIAM J. Num. Anal. 42, 26712687, 2005 MR 2139410 (2006b:82058)
 [10]
 P. Degond, S. Jin and L. Mieussens, A smooth transition model between kinetic and hydrodynamic equations, J. Comp. Phys. 209, 665694, 2005. MR 2151999 (2006g:82037)
 [11]
 P. Degond, J.G. Liu and L. Mieussens, Macroscopic fluid modes with localized kinetic upscaling effects Multiscale Model. Simul. 5, 6951043, 2006. MR 2272306 (2007k:76124)
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 P. Degond, C. Schmeiser, Kinetic boundary layers and fluidkinetic coupling in semiconductors, Transport Theory Statist. Phys. 28, no. 1, 3155, 1999. MR 1669742 (2000a:82064)
 [13]
 F. Golse, S. Jin, C.D. Levermore, A domain decomposition analysis for a twoscale linear transport problem, Math. Model Num. Anal. 37, no. 6, 869892, 2003. MR 2026400 (2004i:65139)
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 S. Jin, Z.P. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48, no. 3, 235276, 1995. MR 1322811 (96c:65134)
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 [17]
 A. Klar, Convergence of alternating domain decomposition schemes for kinetic and aerodynamic equations, Math. Methods Appl. Sci.18, no. 8, 649670, 1995. MR 1335825 (96d:65159)
 [18]
 A. Klar, H. Neunzert, J. Struckmeier, Transition from kinetic theory to macroscopic fluid equations: a problem for domain decomposition and a source for new algorithm, Transp. Theory and Stat. Phys. 29, 93106, 2000.
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 Z.P. Xin, W.Q. Xu, Stiff wellposedness and asymptotic convergence for a class of linear relaxation systems in a quarter plane, Journal of Differential Equations 167, 388437, 2000. MR 1793199 (2001j:35185)
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 Z.P. Xin, W.Q. Xu, Initialboundary value problem to systems of conservation laws with relaxation, Quarterly of applied mathematics 60, no. 2, 251281, 2002. MR 1900493 (2003f:35199)
 [35]
 W.Q. Xu, Boundary conditions for multidimensional hyperbolic relaxation problems, Discrete Contin. Dyn. Syst., 916925, 2003. MR 2018201
 [36]
 X. Yang, F. Golse, Z.Y. Huang, S. Jin, Numerical study of a domain decomposition method for a twoscale linear transport equation, Netw. Heterog. Media 1, no. 1, 143166, 2006. MR 2219280 (2006m:65210)
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Additional Information
Shi Jin
Affiliation:
Department of Mathematics, University of WisconsinMadison, 480 Lincoln Drive, Madison, Wisconsin 53706
Email:
jin@math.wisc.edu
Jianguo Liu
Affiliation:
Department of Physics and Department of Mathematics, Duke University, Durham, North Carolina 27708
Email:
JianGuo.Liu@duke.edu
Li Wang
Affiliation:
Department of Mathematics, University of WisconsinMadison, 480 Lincoln Drive, Madison, Wisconsin 53706
Email:
wangli@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S002557182012026433
Received by editor(s):
February 17, 2011
Received by editor(s) in revised form:
October 9, 2011
Published electronically:
October 9, 2012
Additional Notes:
This research was partially supported by NSF grant No. DMS0608720, and NSF FRG grant DMS0757285. The first author was also supported by a Van Vleck Distinguished Research Prize and a Vilas Associate Award from the University of WisconsinMadison
The second author research was supported by NSF grant DMS 1011738
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
