A Note on the Unified Coordinate System for Computing Shock Waves

In 1999 [J. Comput. Phys.153, 596], Hui and his co-workers proposed a unified coordinate system for computing compressible flows with discontinuous solutions. In their coordinate system, there is a free parameter h such that the traditional Eulerian approach and Lagrangian approach correspond to the particular cases h=0 and h=1, respectively. Hence this approach unifies the two classical methods for describing fluid flows. In this note we consider a one-dimensional problem and we show that there is a parameter range within 0<h<1, such that the coordinate transformation is not invertible across a shock wave. In addition, there is a value of h such that the transformation becomes singular. Hence the parameter h should be restricted to a value close to 0 or 1 near a shock wave. This restriction does not occur away from a shock wave. This note clearly shows that the unified coordinate system of Hui et al. involves interesting properties that should be considered in practical applications.