The Riemann Problem for a Macroscopic Production Model with Logarithmic Equation of State

In this paper, we study the Riemann problem for a second-order macroscopic production model with the anticipated factor of logarithmic state and obtain the expression of the Riemann solution. Interestingly, we find two new types of composite waves. One of them is a composite wave consisting of a rarefaction wave and a double-contact discontinuity attached to the wavefront of the rarefaction wave, where the rarefaction wave and the double-contact discontinuity are waves corresponding to one genuinely nonlinear eigenvalue and the double-contact discontinuity satisfies \(\lambda _1(u_+,\rho _+)=\lambda _1(u _-,\rho _-)\) ${\lambda }_1(u_{+},{\rho }_{+})={\lambda }_1(u_{-},{\rho }_{-})$ . The other is a composite wave consisting of a rarefaction wave and a \(\delta \) $\delta$ standing wave. Subsequently, the stability of the composite wave solutions is proved by perturbing the Riemann initial values. Finally, some representative examples of numerical simulations are given.