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Convergence of MUSCL type methods for scalar conservation laws. (English. Abridged French version) Zbl 0712.65082
The authors discuss the convergence of second order, total variation diminishing, finite difference approximations to the resolvent equations for a scalar, one-dimensional conservation law. Effective schemes should give good resolution of the shocks and contact discontinuities and an extensive summary of previous schemes is given.
The equation considered is essentially of the form \(u_ t+\lambda f(u_ x)=q\) in \({\mathbb{R}}\). The discretized problem \((x_ j=j\Delta x)\) is taken to have the form \(u_ j+\lambda F(D_ -u_ j+\Delta x/2B^-_ jD_+u_ j-\Delta x/2B^+_ j)=g_ k,\) where F is a given numerical flux which is continuous on \({\mathbb{R}}\times {\mathbb{R}}\), \(D_ -u_ j=(u_ j-u_{j-1})/\Delta x\), \(D_+u_ j=(u_{j+1}-u_ j)/\Delta x\) and \(B^{\pm}\) depends on the \(D_{\pm}u_ j\) terms.
On the assumption of a strictly convex flux F, the authors prove convergence towards the entropic weak solution.
Reviewer: A.Swift

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76M20 Finite difference methods applied to problems in fluid mechanics
76M30 Variational methods applied to problems in fluid mechanics
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