Recent developments of the GRP method

A review of about a decade of development of the generalized Riemann problem (GRP) scheme is proposed. Basically, the GRP is an 'analytic high-resolution' (second-order) extension of the classical Godunov scheme, designed to solve numerically systems of conservation or 'quasi-conservation' laws. One can also describe the method as a sort of 'hybrid' scheme, incorporating the detailed analysis of the characteristic structure at singularities into a robust 'shock capturing' method, based on conservative differencing. The various numerical and physical extensions are presented. The range of versatile applications of the GRP method is illustrated through numerous examples.