Thermodynamic relaxation method: Relaxation kernel and predictor equation of state

The use of complex equations of state (EOSs) in compressible flow solvers presents significant theoretical and numerical challenges. These EOSs often present a limited range of validity, leading to “loss-of-convexity” issues, affecting the computed thermodynamic state. Additionally, the complexity of these formulations increases computational cost, as iterative numerical procedures are required, thereby reducing robustness. A novel thermodynamic framework has recently been developed to address these issues. This method, known as the “thermodynamic relaxation method (TRM),” is relevant due to its simplicity, efficiency, and robustness. The method is based on two EOSs: a target equation of state, which represents the physical equation of state, and a predictor equation of state (PEOS), which is simple and convex over a wide range of thermodynamical states. The PEOS is used in the numerical method to close the corresponding hyperbolic system of equations, while thermodynamic parameters are relaxed after each time step to map the solution to the physical target equation. This approach results in faster and more efficient computations. However, the relaxation step, as developed in previous works, is based on heuristic relations, leaving opportunities to strengthen the mathematical foundation of the method. This paper aims to refine the theoretical basis of the TRM and to propose a novel and enhanced relaxation kernel. Additionally, the refinement of the PEOS is discussed. The method also yields a more accurate and consistent thermal equilibrium temperature.