Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

John M. Hong, Ying Chin Su

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4 Scopus citations


In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by "weaker than weak" solutions of the Riemann problem, J. Differential Equations 222 (2006) 515-549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697-715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.

Original languageEnglish
Pages (from-to)635-650
Number of pages16
JournalNonlinear Analysis, Theory, Methods and Applications
Issue number2
StatePublished - 15 Jan 2009


  • Boundary Riemann problem
  • Generalized Glimm scheme
  • Hyperbolic systems of balance laws
  • Lax's method
  • Riemann problem


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