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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Error bounds for Glimm difference approximations for scalar conservation laws


Authors: David Hoff and Joel Smoller
Journal: Trans. Amer. Math. Soc. 289 (1985), 611-642
MSC: Primary 65M15; Secondary 35L65, 76L05
MathSciNet review: 784006
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Abstract: We derive error bounds for the Glimm difference approximation to the solution of a genuinely nonlinear scalar conservation law with BV initial data. We show that the $ {L^1}$ error is bounded by $ O(\Delta {x^{1/6}}\vert\log \Delta x\vert)$ in the general case, and by $ O(\Delta {x^{1/2}}\vert\log \Delta x\vert)$ for a generic class of piecewise constant data.


References [Enhancements On Off] (What's this?)

  • [G] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 0194770 (33 #2976)
  • [GL] James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767 (42 #676)
  • [K] S. N. Krushkov, First order quasilinear equations in several space variables, Math. USSR Sb. 10 (1970), 217-273.
  • [Ku N] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0419394 (54 #7415)
  • [Ku] N. N. Kuznetsov, On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 183–197. MR 0657786 (58 #31874)
  • [L] Tai Ping Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), no. 2, 135–148. MR 0470508 (57 #10259)
  • [S] Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146 (84d:35002)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0784006-5
PII: S 0002-9947(1985)0784006-5
Article copyright: © Copyright 1985 American Mathematical Society