Holden, Helge; Holden, Lars; Høegh-Krohn, Raphael A numerical method for first order nonlinear scalar conservation laws in one-dimension. (English) Zbl 0658.65085 Comput. Math. Appl. 15, No. 6-8, 595-602 (1988). A numerical method for the following initial value problem for first order nonlinear scalar hyperbolic conservation law \(u_ t+f(u)_ x=0,\) \(u(x,0)=u_ 0(x)\) is proposed. The method is to approximate f by a piecewise linear function and the initial value by a piecewise constant function. It is proved to be applicable as a numerical method for a general flux function and a general initial value. The authors also point out that the error in this method is far smaller than in any other method and illustrate the method in an example. Reviewer: V.Kamen Cited in 26 Documents MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems 35L65 Hyperbolic conservation laws Keywords:finite method; numerical example; first order nonlinear scalar hyperbolic conservation law; flux function Citations:Zbl 0233.35014 PDFBibTeX XMLCite \textit{H. Holden} et al., Comput. Math. Appl. 15, No. 6--8, 595--602 (1988; Zbl 0658.65085) Full Text: DOI References: [1] Oleinik, O. A., Am. math. Soc. Transl. Ser., 2, 26, 95-172 (1963), Also · Zbl 0131.31803 [2] Oleinik, O. A., Am. math. Soc. Transl. Ser., 2, 33, 285-290 (1964), Also · Zbl 0132.33303 [3] Volpert, A. I., The Spaces BV and quasilinear equations, Math. USSR Sb., 2, 225-267 (1967) · Zbl 0168.07402 [4] Kruzkov, S. N., First order quasilinear equations in several independent variables, Math. USSR Sb., 10, 217-243 (1970) · Zbl 0215.16203 [5] Lax, P. D., Weak solutions of hyperbolic equations and their numerical computation, Communs pure appl. Math., 7, 159-193 (1954) · Zbl 0055.19404 [6] Dafermos, C. M., Polygonal approximation of solutions of the initial value problem for a conservation law, J. math. Analysis Applic., 38, 33-41 (1972) · Zbl 0233.35014 [7] Lucier, L. J., A moving mesh numerical method for hyperbolic conservation laws, Math. Comput., 46, 59-69 (1986) · Zbl 0592.65062 [8] LeVeque, R. J., A large time step shock-capturing techniques for scalar conservation laws, SIAM Jl Numer. Analysis, 19, 1051-1073 (1982) [9] Lucier, L. J., Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM Jl Numer. Analysis, 22, 1074-1081 (1985) · Zbl 0584.65059 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.