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On nonlocal monotone difference schemes for scalar conservation laws. (English) Zbl 0604.65061
An analysis of the error estimates for the approximate solution of the (scalar) hyperbolic conservation law \(u_ t+f(u)_ x=0\), \(X\in {\mathbb{R}}\), \(0<t\leq T\), \(u(x,0)=u_ 0(x)\), \(X\in {\mathbb{R}}\) is given. The numerical mesh proposed is the uniform one rather than that using that suggested by the characteristic coordinates, though consideration is given to looking at finite element methods. Convergence results are given for the error estimates for the corresponding explicit, implicit, and semi-infinite finite-difference schemes. Stability analyses are given when a dissipative term \((-\nu g(u)_{xx})\) and a dispersive term \((- \alpha^ 2 u_{xxt})\) is added to the equation above.
The estimates are extended to the weak solutions of the equation thereby.
Reviewer: G.Wake

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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