An effective all-speed Riemann solver with self-similar internal structure for Euler system

The HLLI Riemann solver proposed by Balsara is a universal scheme which is capable of resolving full family of intermediate waves with minimal dissipation for hyperbolic conservation law. However, asymptotic analyses demonstrate that such a scheme is incapable of obtaining physical solutions at low speeds due to excessive numerical dissipations of the velocity-difference terms in the momentum equations. To accommodate this deficiency, we employ the construction of the HLLEM scheme to separate shear waves which contain velocity difference in the tangential direction from contact waves and modify the quantity of the velocity difference. In addition, a function κ is adopted to damp the shear waves in the vicinity of the shockwaves to avoid shock instability phenomena. With these improvements, proper dissipation can be obtained at all speeds without suffering from the appearance of shock anomalies. Finally, a series of numerical experiments illustrate that the novel HLLIM scheme we propose is effective and robust at all speeds. Therefore, analyses in this study could provide reference for the improvement of the HLLI solver for the other hyperbolic conservation law in cases the eigenvalues vary sharply.