Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state

For the numerical simulation of detonation of condensed phase explosives, a complex equation of state (EOS), such as the Jones–Wilkins–Lee (JWL) EOS or the Cochran–Chan (C–C) EOS, are widely used. However, when a conservative scheme is used for solving the Euler equations with such equations of state, a spurious solution across the contact discontinuity, a well known phenomenon in multi-fluid systems, arises even for single materials. In this work, we develop a generalised Osher-type scheme in an adaptive primitive–conservative framework to overcome the aforementioned difficulties. Resulting numerical solutions are compared with the exact solutions and with the numerical solutions from the Godunov method in conjunction with the exact Riemann solver for the Euler equations with Mie–Grüneisen form of equations of state, such as the JWL and the C–C equations of state. The adaptive scheme is extended to second order and its empirical convergence rates are presented, verifying second order accuracy for smooth solutions. Through a suite of several tests problems in one and two space dimensions we illustrate the failure of conservative schemes and the capability of the methods of this paper to overcome the difficulties.