A Variant of Van Leer's Method for Multidimensional Systems of Conservation Laws

We present a new variant of Van Leer's construction of upwind finite volume schemes for hyperbolic systems of conservation laws. Fluxes are computed with second-order accuracy using an interpolation rather than a slope reconstruction. A correction of the interpolated values is necessary and performed globally on each cell by a conservation argument. It can be used on a rectangular or triangles based dual grid to obtain a genuinely multidimensional scheme. One of our main concerns in this construction, is to prove that the second-order reconstruction, combined with a Boltzmann solver, gives nonnegative values of the pressure and density for gas dynamics, even on an unstructured mesh. This allows us to derive a rigorous CFL condition. Thus our approach is very robust.