On Godunov-Type Schemes for Lagrangian Gas Dynamics

In this paper, Godunov-type schemes are considered for the equations of gas dynamics using Lagrangian coordinates. A Roe linearization is constructed for a general equation of state. It does not coincide with that for Eulerian coordinates. It is shown that this linearization fails in the vicinity of strong compressions, in the sense that the approximate Riemann solution contains unphysical states of negative specific volume. An algorithm to calculate a priori bounds for the smallest and largest signal velocity is obtained by correcting the signal velocities of this Roe linearization. These bounds are used within a very simple Godunov-type scheme which captures strong compressions very well. Numerical results are shown for several test problems.