Remark on the generalized Riemann problem method for compressible fluid flows

The generalized Riemann problem (GRP) method was proposed for compressible fluid flows based on the Lagrangian formulation [M. Ben-Artzi, J. Falcovitz, A second-order Godunov-type scheme for compressible fluid dynamics, J. Comput. Phys., 55(1) (1984) 1–32], and a direct Eulerian version was developed in [M. Ben-Artzi, J. Li, G. Warnecke, A direct Eulerian GRP scheme for compressible fluid flows, J. Comput. Phys., 28 (2006) 19–43] by using the concept of Riemann invariants. The central feature of the GRP method is the resolution of centered rarefaction waves. In this note we show how to use the concept of Riemann invariants in order to resolve the rarefaction waves in the Lagrangian coordinate system and result in the GRP scheme.