CONVERGENCE OF THE FINITE VOLUME METHOD
    FOR MULTIDIMENSIONAL CONSERVATION LAWS

           B. Cockburn, F. Coquel, and P.G. LeFloch

abstract
 We establish the convergence of the finite
 volume method applied to multidimensional hyperbolic conservation laws, 
and based on monotone numerical flux-functions. 
Our technique applies under a fairly unrestrictive assumption on the triangulations 
(``flat elements'' are allowed),
 and to Lipschitz continuous flux-functions. 
 We treat the initial and boundary value problem, and obtain
 the strong convergence of the scheme to the unique entropy discontinuous solution in 
the sense of Kruzkov. The proof of convergence is based on a convergence framework  
due to Coquel and LeFloch (Math. of Comp. 57 (1991), 169--210),      
& J. Numer. Anal.  30 (1993), 675--700.  
  From a convex decomposition of the scheme,
we derive a new estimate for the rate of entropy dissipation, 
and a new formulation of the discrete entropy inequalities. These estimates are shown to 
be sufficient for the passage to the limit in the discrete equation. 
Convergence follows from DiPerna's uniqueness result
in the class of entropy measure-valued solutions.

 The whole paper is available as a compressed Postscript file by internet
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       cockburn_al.290.nov.93.ps.gz