Localization effects and measure source terms in numerical schemes for balance laws

Gosse, Laurent

doi:10.1090/S0025-5718-01-01354-0

Localization effects and measure source terms in numerical schemes for balance laws
HTML articles powered by AMS MathViewer

by Laurent Gosse PDF

Math. Comp. 71 (2002), 553-582 Request permission

Abstract:

This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided.

References

D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal., 61 (1996), pp. 163–193.
Paolo Baiti and Helge Kristian Jenssen, Well-posedness for a class of $2\times 2$ conservation laws with $L^\infty$ data, J. Differential Equations 140 (1997), no. 1, 161–185. MR 1473859, DOI 10.1006/jdeq.1997.3308
C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017–1034. MR 542510, DOI 10.1080/03605307908820117
F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys. 95 (1999), no. 1-2, 113–170. MR 1705583, DOI 10.1023/A:1004525427365
F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32 (1998), no. 7, 891–933. MR 1618393, DOI 10.1016/S0362-546X(97)00536-1
F. Bouchut and B. Perthame, Kružkov’s estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2847–2870 (English, with English and French summaries). MR 1475677, DOI 10.1090/S0002-9947-98-02204-1
P. A. Burton, Convergence of flux-limiting schemes for hyperbolic conservation laws with source terms, Ph.D. Thesis, Univ. of Reading (1993), available at the following URL: http://www.rdg.ac.uk/AcadDepts/sm/wsm1/theses/pab.html.
Patricia Cargo and Alain Yves LeRoux, Un schéma équilibre adapté au modèle d’atmosphère avec termes de gravité, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 1, 73–76 (French, with English and French summaries). MR 1260539
J.-J. Cauret, J.-F. Colombeau, and A. Y. LeRoux, Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations, J. Math. Anal. Appl. 139 (1989), no. 2, 552–573. MR 996979, DOI 10.1016/0022-247X(89)90129-7
Michael G. Crandall and Andrew Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), no. 149, 1–21. MR 551288, DOI 10.1090/S0025-5718-1980-0551288-3
C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), no. 6, 1097–1119. MR 457947, DOI 10.1512/iumj.1977.26.26088
Gianni Dal Maso, Philippe G. Lefloch, and François Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9) 74 (1995), no. 6, 483–548. MR 1365258
Pedro Embid, Jonathan Goodman, and Andrew Majda, Multiple steady states for $1$-D transonic flow, SIAM J. Sci. Statist. Comput. 5 (1984), no. 1, 21–41. MR 731879, DOI 10.1137/0905002
Bjorn Engquist and Björn Sjögreen, The convergence rate of finite difference schemes in the presence of shocks, SIAM J. Numer. Anal. 35 (1998), no. 6, 2464–2485. MR 1655855, DOI 10.1137/S0036142997317584
J. Falcovitz and M. Ben-Artzi, Recent developments of the GRP method, JSME International J. Series B, 38 (1995), pp. 497–517.
James Glimm, Guillermo Marshall, and Bradley Plohr, A generalized Riemann problem for quasi-one-dimensional gas flows, Adv. in Appl. Math. 5 (1984), no. 1, 1–30. MR 736548, DOI 10.1016/0196-8858(84)90002-2
Laurent Gosse, A priori error estimate for a well-balanced scheme designed for inhomogeneous scalar conservation laws, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 5, 467–472 (English, with English and French summaries). MR 1652550, DOI 10.1016/S0764-4442(99)80024-X
Laurent Gosse, Sur la stabilité des approximations implicites des lois de conservation scalaires non homogènes, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 79–84 (French, with English and French summaries). MR 1703322, DOI 10.1016/S0764-4442(99)80466-2
L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms, Math. Mod. Meth. Appl. Sc., 11 (2001), pp. 339–365.
L. Gosse, A well-balanced flux splitting scheme designed for hyperbolic systems of conservation laws with source terms, Comp. Math. Applic. 39 (2000), pp. 135–159.
Laurent Gosse and François James, Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients, Math. Comp. 69 (2000), no. 231, 987–1015. MR 1670896, DOI 10.1090/S0025-5718-00-01185-6
Laurent Gosse and Alain-Yves Leroux, Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 5, 543–546 (French, with English and French summaries). MR 1408992
J. M. Greenberg and A. Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996), no. 1, 1–16. MR 1377240, DOI 10.1137/0733001
Jean-François Colombeau and Arnaud Heibig, Nonconservative products in bounded variation functions, SIAM J. Math. Anal. 23 (1992), no. 4, 941–949. MR 1166566, DOI 10.1137/0523050
Christiane Helzel, Randall J. Leveque, and Gerald Warnecke, A modified fractional step method for the accurate approximation of detonation waves, SIAM J. Sci. Comput. 22 (2000), no. 4, 1489–1510. MR 1796622, DOI 10.1137/S1064827599357814
Thomas Y. Hou and Philippe G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis, Math. Comp. 62 (1994), no. 206, 497–530. MR 1201068, DOI 10.1090/S0025-5718-1994-1201068-0
Eli Isaacson and Blake Temple, Convergence of the $2\times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math. 55 (1995), no. 3, 625–640. MR 1331577, DOI 10.1137/S0036139992240711
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, DOI 10.1002/cpa.3160480303
Markos A. Katsoulakis and Athanasios E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations 22 (1997), no. 1-2, 195–233. MR 1434144, DOI 10.1080/03605309708821261
Runhild Aae Klausen and Nils Henrik Risebro, Stability of conservation laws with discontinuous coefficients, J. Differential Equations 157 (1999), no. 1, 41–60. MR 1710013, DOI 10.1006/jdeq.1998.3624
S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
Philippe G. Lefloch and Athanasios E. Tzavaras, Representation of weak limits and definition of nonconservative products, SIAM J. Math. Anal. 30 (1999), no. 6, 1309–1342. MR 1718304, DOI 10.1137/S0036141098341794
R. J. LeVeque and H. C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys. 86 (1990), no. 1, 187–210. MR 1033905, DOI 10.1016/0021-9991(90)90097-K
Tai-Ping Liu, Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 28 (1987), no. 11, 2593–2602. MR 913412, DOI 10.1063/1.527751
Corrado Mascia and Andrea Terracina, Large-time behavior for conservation laws with source in a bounded domain, J. Differential Equations 159 (1999), no. 2, 485–514. MR 1730729, DOI 10.1006/jdeq.1999.3669
J. P. Raymond, A new definition of nonconservative products and weak stability results, Boll. Un. Mat. Ital. B (7) 10 (1996), no. 3, 681–699 (English, with Italian summary). MR 1411523
T. Tang and Zhen Huan Teng, Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal. 32 (1995), no. 1, 110–127. MR 1313707, DOI 10.1137/0732004
Blake Temple, Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc. 280 (1983), no. 2, 781–795. MR 716850, DOI 10.1090/S0002-9947-1983-0716850-2
Bram van Leer, On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe, SIAM J. Sci. Statist. Comput. 5 (1984), no. 1, 1–20. MR 731878, DOI 10.1137/0905001
Alexis Vasseur, Kinetic semidiscretization of scalar conservation laws and convergence by using averaging lemmas, SIAM J. Numer. Anal. 36 (1999), no. 2, 465–474. MR 1668278, DOI 10.1137/S0036142996313610
A. I. Vol′pert, Spaces $\textrm {BV}$ and quasilinear equations, Mat. Sb. (N.S.) 73 (115) (1967), 255–302 (Russian). MR 0216338