The Riemann problem for the one-dimensional compressible flow of a van der Waals gas

We address the Riemann problem for the one-dimensional compressive flow of a van der Waals gas, in which the external force is a given continuous function of time. The explicit forms for shock wave, contact discontinuity and rarefaction wave are presented. The influence of van der Waals parameter on these elementary waves is discussed. Furthermore, we obtain six kinds of solutions to this Riemann problem and establish a condition for the occurrence of vacuum in the solutions. Particularly, all of the solutions are not self-similar in (t, x)-plane due to the presence of the time-dependent external force.