Error bounds for Glimm difference approximations for scalar conservation laws. (English) Zbl 0579.65096

This paper treats the Glimm difference approximation to the solution of a genuinely nonlinear scalar conservation law \(v_ t+f(v)_ x==0\), \((x,t)\in R\times R_+\), with initial data \(v(x,0)=v_ o(x)\) with bounded variation. Error bounds for this problem are derived in the general case and for the generic case of piecewise constant initial data.
Reviewer: V.Drápalík


65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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[1] James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697 – 715. · Zbl 0141.28902
[2] 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. · Zbl 0204.11304
[3] S. N. Krushkov, First order quasilinear equations in several space variables, Math. USSR Sb. 10 (1970), 217-273.
[4] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. · Zbl 0281.10001
[5] 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.
[6] Tai Ping Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), no. 2, 135 – 148. · Zbl 0376.35042
[7] 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. · Zbl 0508.35002
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