Positively conservative high-resolution convection schemes for unstructured elements

Despite their geometric flexibility, unstructured mesh schemes for compressible gas dynamics do not usually resolve captured shocks and contact discontinuities as well as corresponding structured mesh schemes. The main reason for this appears to be the difficulty in constructing analogous extensions to higher-order accuracy. This issue is addressed in some detail and a new, compact stencil, Maximum Limited Gradient (MLG) reconstruction technique is presented for unstructured elements. The MLG reconstruction turns out to be a multidimensional analogue of the one-dimensional Superbee slope. We then describe a simple and robust extension to systems of equations, which does not require any diagonalization of flux Jacobian matrices. An application to a blast wave hazard prediction problem is presented using the wave-by-wave extension of the MLG limiter to the Euler equations.