The vanishing pressure limits in Riemann solutions for the non-isentropic Euler equations with combined pressure and time-dependent source term

In this paper, we investigate the vanishing pressure limits of Riemann solutions for the one-dimensional (1-D) non-isentropic Euler equations with a time-dependent source term and a combined equation of state incorporating both polytropic and logarithmic terms. Since the right-hand side of equations contains a time-varying source term, the solutions of the Riemann problem lose self-similarity. To address this challenge, a velocity transformation is employed to convert the 1-D Euler system with a time-dependent source term into the conservation law form. Firstly, based on the characteristic analysis and phase plane analysis, Riemann solutions are obtained for the transformed conservative system, encompassing rarefaction wave, contact discontinuity, and shock wave. Subsequently, the Riemann solutions of the original system are derived by the velocity transformation. Furthermore, the limiting behavior of these Riemann solutions are studied as the pressure tends to zero. The analytical results demonstrate that the Riemann solutions of the original system, which consist of a 1-shock wave, a 3-shock wave, and a contact discontinuity, converge to the delta shock (δ-shock) wave solution of the pressureless Euler system with the same source term. The solutions composed of a 1-rarefaction wave, a 3-rarefaction wave, and a contact discontinuity, converge to the pressureless Euler system with a source term, involving the contact discontinuity and vacuum. Finally, some numerical examples are provided to show the formation of δ-shock wave and vacuum.