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.
 [1]
Guillaume
Bal and Yvon
Maday, Coupling of transport and diffusion models in linear
transport theory, M2AN Math. Model. Numer. Anal. 36
(2002), no. 1, 69–86. MR 1916293
(2003e:82065), 10.1051/m2an:2002007
 [2]
Stefano
Bianchini, Hyperbolic limit of the JinXin relaxation model,
Comm. Pure Appl. Math. 59 (2006), no. 5,
688–753. MR 2172805
(2008b:35167), 10.1002/cpa.20114
 [3]
J.F.
Bourgat, P.
Le Tallec, B.
Perthame, and Y.
Qiu, Coupling Boltzmann and Euler equations without
overlapping, Domain decomposition methods in science and engineering
(Como, 1992), Contemp. Math., vol. 157, Amer. Math. Soc., Providence,
RI, 1994, pp. 377–398. MR 1262639
(95d:76085), 10.1090/conm/157/01439
 [4]
Alberto
Bressan, Hyperbolic systems of conservation laws, Oxford
Lecture Series in Mathematics and its Applications, vol. 20, Oxford
University Press, Oxford, 2000. The onedimensional Cauchy problem. MR 1816648
(2002d:35002)
 [5]
Carlo
Cercignani, The Boltzmann equation and its applications,
Applied Mathematical Sciences, vol. 67, SpringerVerlag, New York,
1988. MR
1313028 (95i:82082)
 [6]
A.
Chalabi and D.
Seghir, Convergence of relaxation schemes for initial boundary
value problems for conservation laws, Comput. Math. Appl.
43 (2002), no. 89, 1079–1093. MR 1892486
(2003a:35124), 10.1016/S08981221(02)800141
 [7]
Gui
Qiang Chen, C.
David Levermore, and TaiPing
Liu, Hyperbolic conservation laws with stiff relaxation terms and
entropy, Comm. Pure Appl. Math. 47 (1994),
no. 6, 787–830. MR 1280989
(95h:35133), 10.1002/cpa.3160470602
 [8]
Pierre
Degond, Giacomo
Dimarco, and Luc
Mieussens, A multiscale kineticfluid solver with dynamic
localization of kinetic effects, J. Comput. Phys. 229
(2010), no. 13, 4907–4933. MR 2643635
(2011d:82089), 10.1016/j.jcp.2010.03.009
 [9]
Pierre
Degond and Shi
Jin, A smooth transition model between kinetic and diffusion
equations, SIAM J. Numer. Anal. 42 (2005),
no. 6, 2671–2687 (electronic). MR 2139410
(2006b:82058), 10.1137/S0036142903430414
 [10]
Pierre
Degond, Shi
Jin, and Luc
Mieussens, A smooth transition model between kinetic and
hydrodynamic equations, J. Comput. Phys. 209 (2005),
no. 2, 665–694. MR 2151999
(2006g:82037), 10.1016/j.jcp.2005.03.025
 [11]
Pierre
Degond, JianGuo
Liu, and Luc
Mieussens, Macroscopic fluid models with localized kinetic
upscaling effects, Multiscale Model. Simul. 5 (2006),
no. 3, 940–979 (electronic). MR 2272306
(2007k:76124), 10.1137/060651574
 [12]
Pierre
Degond and Christian
Schmeiser, Kinetic boundary layers and fluidkinetic coupling in
semiconductors, Transport Theory Statist. Phys. 28
(1999), no. 1, 31–55. MR 1669742
(2000a:82064), 10.1080/00411459908214514
 [13]
François
Golse, Shi
Jin, and C.
David Levermore, A domain decomposition analysis for a twoscale
linear transport problem, M2AN Math. Model. Numer. Anal.
37 (2003), no. 6, 869–892. MR 2026400
(2004i:65139), 10.1051/m2an:2003059
 [14]
Robert
L. Higdon, Initialboundary value problems for linear hyperbolic
systems, SIAM Rev. 28 (1986), no. 2,
177–217. MR
839822 (88a:35138), 10.1137/1028050
 [15]
Shi
Jin and Zhou
Ping Xin, The relaxation schemes for systems of conservation laws
in arbitrary space dimensions, Comm. Pure Appl. Math.
48 (1995), no. 3, 235–276. MR 1322811
(96c:65134), 10.1002/cpa.3160480303
 [16]
HeinzOtto
Kreiss, Initial boundary value problems for hyperbolic
systems, Comm. Pure Appl. Math. 23 (1970),
277–298. MR 0437941
(55 #10862)
 [17]
Axel
Klar, Convergence of alternating domain decomposition schemes for
kinetic and aerodynamic equations, Math. Methods Appl. Sci.
18 (1995), no. 8, 649–670. MR 1335825
(96d:65159), 10.1002/mma.1670180806
 [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.
 [19]
S.
N. Kružkov, First order quasilinear equations with several
independent variables., Mat. Sb. (N.S.) 81 (123)
(1970), 228–255 (Russian). MR 0267257
(42 #2159)
 [20]
L.D. Landau, E.M. Lifschitz, Statistical Physics, Elsevier (Singapore) Pte Ltd, 1980.
 [21]
Randall
J. LeVeque and Zhi
Lin Li, The immersed interface method for elliptic equations with
discontinuous coefficients and singular sources, SIAM J. Numer. Anal.
31 (1994), no. 4, 1019–1044. MR 1286215
(95g:65139), 10.1137/0731054
 [22]
C.
David Levermore, Moment closure hierarchies for kinetic
theories, J. Statist. Phys. 83 (1996), no. 56,
1021–1065. MR 1392419
(97e:82041), 10.1007/BF02179552
 [23]
Andrew
Majda and Stanley
Osher, Initialboundary value problems for hyperbolic equations
with uniformly characteristic boundary, Comm. Pure Appl. Math.
28 (1975), no. 5, 607–675. MR 0410107
(53 #13857)
 [24]
Roberto
Natalini, Convergence to equilibrium for the relaxation
approximations of conservation laws, Comm. Pure Appl. Math.
49 (1996), no. 8, 795–823. MR 1391756
(97c:35131), 10.1002/(SICI)10970312(199608)49:8<795::AIDCPA2>3.0.CO;23
 [25]
Roberto
Natalini, Recent results on hyperbolic relaxation problems,
Analysis of systems of conservation laws (Aachen, 1997) Chapman &
Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 99, Chapman &
Hall/CRC, Boca Raton, FL, 1999, pp. 128–198. MR 1679940
(2000a:35157)
 [26]
Roberto
Natalini and Bernard
Hanouzet, Weakly coupled systems of quasilinear hyperbolic
equations, Differential Integral Equations 9 (1996),
no. 6, 1279–1292. MR 1409928
(97h:35138)
 [27]
James
V. Ralston, Note on a paper of Kreiss, Comm. Pure Appl. Math.
24 (1971), no. 6, 759–762. MR 0606239
(58 #29326)
 [28]
ZhenHuan
Teng, Firstorder 𝐿¹convergence for relaxation
approximations to conservation laws, Comm. Pure Appl. Math.
51 (1998), no. 8, 857–895. MR 1620220
(99f:65133), 10.1002/(SICI)10970312(199808)51:8<857::AIDCPA1>3.3.CO;2M
 [29]
M.
Tidriri, New models for the solution of intermediate regimes in
transport theory and radiative transfer: existence theory, positivity,
asymptotic analysis, and approximations, J. Statist. Phys.
104 (2001), no. 12, 291–325. MR 1851390
(2002j:82104), 10.1023/A:1010365812733
 [30]
W.G. Vincenti, C.H. Kruger, Introduction to Physical Gas Dynamics, Wiley, New York, 1965.
 [31]
WeiCheng
Wang and Zhouping
Xin, Asymptotic limit of initialboundary value problems for
conservation laws with relaxational extensions, Comm. Pure Appl. Math.
51 (1998), no. 5, 505–535. MR 1604274
(99a:35172), 10.1002/(SICI)10970312(199805)51:5<505::AIDCPA3>3.0.CO;2C
 [32]
G.
B. Whitham, Linear and nonlinear waves, WileyInterscience
[John Wiley & Sons], New YorkLondonSydney, 1974. Pure and Applied
Mathematics. MR
0483954 (58 #3905)
 [33]
Zhouping
Xin and WenQing
Xu, Stiff wellposedness and asymptotic convergence for a class of
linear relaxation systems in a quarter plane, J. Differential
Equations 167 (2000), no. 2, 388–437. MR 1793199
(2001j:35185), 10.1006/jdeq.2000.3806
 [34]
Zhouping
Xin and WenQing
Xu, Initialboundary value problem to systems of conservation laws
with relaxation, Quart. Appl. Math. 60 (2002),
no. 2, 251–281. MR 1900493
(2003f:35199)
 [35]
WenQing
Xu, Boundary conditions for multidimensional hyperbolic relaxation
problems, Discrete Contin. Dyn. Syst. suppl. (2003),
916–925. Dynamical systems and differential equations (Wilmington,
NC, 2002). MR
2018201
 [36]
Xu
Yang, Francois
Golse, Zhongyi
Huang, and Shi
Jin, Numerical study of a domain decomposition method for a
twoscale linear transport equation, Netw. Heterog. Media
1 (2006), no. 1, 143–166. MR 2219280
(2006m:65210), 10.3934/nhm.2006.1.143
 [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)
 [8]
 P. Degond, G. Dimarco and L. Mieussens, A multiscale kineticfluid solver with dynamic localization of kinetic effects, J. Comput. Phys., 229, 49074933, 2010. MR 2643635 (2011d:82089)
 [9]
 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)
 [12]
 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)
 [14]
 R.L. Higdon, Initialboundary value problems for linear hyperbolic systems, SIAM Review, vol. 28, no. 2, 177217, 1986. MR 839822 (88a:35138)
 [15]
 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)
 [16]
 H.O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math. 23, 277298, 1970. MR 0437941 (55:10862)
 [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.
 [19]
 S.N. Kruzkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81(123), 228255, 1970. MR 0267257 (42:2159)
 [20]
 L.D. Landau, E.M. Lifschitz, Statistical Physics, Elsevier (Singapore) Pte Ltd, 1980.
 [21]
 R.J. Leveque, Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31, 10191044, 1994 MR 1286215 (95g:65139)
 [22]
 C.D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys. 83, no. 56, 10211065, 1996. MR 1392419 (97e:82041)
 [23]
 A. Majda, S. Osher, Initialboundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math. 28, no. 78, pp. 607675, 1975. MR 0410107 (53:13857)
 [24]
 R. Natalini, Convergence to equilibrium for the relaxation approximation of conservation laws, Comm. Pure Appl. Math. 49, no. 8, 795823, 1996. MR 1391756 (97c:35131)
 [25]
 R. Natalini, Recent mathematical results on hyperbolic relaxation problems, Analysis of Systems of Conservation Laws (Aachen 1997), Chapman Hall/CRC, Boca Raton, 128198, 1999. MR 1679940 (2000a:35157)
 [26]
 R. Natalini, B. Hanouzet, Weakly coupled system of quasilinear hyperbolic equations, Differential Integral Equations 9, no. 6, 12791292, 1996. MR 1409928 (97h:35138)
 [27]
 J.V. Ralston, Note on a paper of Kreiss, Comm. Pure Appl. Math. 24, pp. 759762, 1971. MR 0606239 (58:29326)
 [28]
 Z.H. Teng, Firstorder convergence for relaxation approximations to conservation laws, Comm. Pure Appl. Math., Vol. LI, 08750895, 1998. MR 1620220 (99f:65133)
 [29]
 M. Tidriri, New models for the solution of intermediate regimes in transport theory and radiative transfer: existence theory, positivity, asymptotic analysis, and approximations, J. Stat. Phys. 104, 291325, 2001. MR 1851390 (2002j:82104)
 [30]
 W.G. Vincenti, C.H. Kruger, Introduction to Physical Gas Dynamics, Wiley, New York, 1965.
 [31]
 W.C. Wang, Z.P. Xin, Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions, Comm. Pure Appl. Math., Vol. LI, 05050535, 1998. MR 1604274 (99a:35172)
 [32]
 G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. MR 0483954 (58:3905)
 [33]
 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)
 [34]
 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
PII:
S 00255718(2012)026433
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.
