Numerical viscosity and convergence of finite volume methods for conservation laws with boundary conditions

The authors study the convergence of finite volume schemes for general multidimensional conservation laws with boundary conditions. A unique result for a measure-valued solution, which generalizes those of Diperna for unbounded domains and Szepessy for bounded domains, is proved. It gives us sufficient conditions to get convergence. By studying carefully the entropy production for one-dimensional E-schemes we are able to prove convergence of finite volume E-schemes under general assumption.