CONVERGENCE OF AN UPSTREAM FINITE-VOLUME SCHEME FOR A NONLINEAR HYPERBOLIC EQUATION ON A TRIANGULAR MESH

We study here the discretisation of the nonlinear hyperbolic equation u(t) + div(vf(u)) = 0 in R2 x R+, with given initial condition u(., 0) = u0(.) in R2, where v is a function from R2 X R+ to R2 such that divv = 0 and f is a given nondecreasing function from R to R. An explicit Euler scheme is used for the time discretisation of the equation, and a triangular mesh for the spatial discretisation. Under a usual stability condition, we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation.