A second-order Godunov-type scheme for compressible fluid dynamics

A second-order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented. Two related versions are proposed, one Lagrangian and the second direct Eulerian. They both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity, assuming initial nonvanishing slopes on both sides. While this solution is an extension of the solution to the classical Riemann problem, the resulting schemes are second-order extensions of Godunov's methods. In both cases, they are very simple to implement in computer codes. Several numerical examples are shown, where the only additional mechanism is a simple monotonicity algorithm.