A unified coordinate system for solving the two-dimensional Euler equations
For two-dimensional gas dynamical Euler equations, the authors introduce a unified coordinate system in which the flow variables are functions of time and of some permanent identification of pseudo-particles which move with velocity \(hq\), \(q\) being the velocity of fluid particles. This coordinate system includes Eulerian coordinates as a special case when \(h=0\), and Lagrangian ones when \(h=1\). The two-dimensional unsteady Euler equations in the unified coordinates are found to be hyperbolic for all values of \(h\), except when \(h=1\) (i.e. in Lagrangian coordinates). In the latter case Euler equations are only weakly hyperbolic, lacking one eigenvector, although all eigenvalues are real. The authors discuss the consequences of this deficiency of Lagrangian coordinates in connection with numerical computations.
two-dimensional gas dynamical Euler equations
scientific article; zbMATH DE number 1384964