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A numerical method for first order nonlinear scalar conservation laws in one-dimension. (English) Zbl 0658.65085
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

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
Full Text: DOI
[1] Oleinik, O.A.; Oleinik, O.A., Discountinuous solutions of non-linear differential equations, Usp. mat. nauk., Am. math. soc. transl. ser., 2, 26, 95-172, (1963), Also · Zbl 0131.31803
[2] Oleinik, O.A.; Oleinik, O.A., Uniqueness and a stability of the generalized solution of the chauchy problem for a quasilinear equation, Usp. mat. nauk., 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
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