On local minimum entropy principle of high-order schemes for relativistic Euler equations

The minimum entropy principle (MEP), first established by E. Tadmor [Appl. Numer. Math. 2 (1986), pp. 211–219] for the nonrelativistic Euler system, states that the minimum of the initial specific entropy serves as a lower bound for the specific entropy at all future times. This fundamental principle provides the best-known a priori estimate for entropy in the (nonrelativistic) Euler equations and has been successfully incorporated into the design of stable numerical schemes (see, e.g., B. Khobalatte and B. Perthame [Math. Comp. 62 (1994), pp. 119–131]; X. Zhang and C.-W. Shu [Numer. Math. 121 (2012), pp. 545–563]). However, compared to the nonrelativistic case, the understanding of entropy in relativistic Euler equations remains far more limited, due to the complexities of nonlinear entropy and the intricate mathematical structure of the relativistic Euler system. This paper first establishes the MEP for the relativistic Euler equations with a broad class of general equations of state (EOSs) that satisfy relativistic causality. Furthermore, we address the challenge of preserving the local version of the discovered MEP in high-order numerical schemes. At the continuous level, we find out a family of entropy pairs for the relativistic Euler equations with a general EOS and provide rigorous analysis to prove the strict convexity of entropy under a necessary and sufficient condition. At the numerical level, we develop a rigorous framework for designing provably entropy-preserving high-order schemes that ensure both physical admissibility and the discovered MEP. The relativistic effects, coupled with the abstract and general EOS formulation, introduce significant challenges not encountered in the nonrelativistic case or with the ideal EOS. In particular, entropy is a highly nonlinear and implicit function of the conservative variables, making it particularly difficult to enforce entropy preservation. To address these challenges, we establish a series of auxiliary theories via highly technical inequalities. Another key innovation is the use of geometric quasi-linearization, which reformulates the nonlinear constraints into equivalent linear ones by introducing additional free parameters. These advancements form the foundation of our entropy-preserving analysis. We propose novel, robust, locally entropy-preserving high-order frameworks. A central challenge is accurately estimating the local minimum of entropy, particularly in the presence of shock waves at unknown locations. To address this, we introduce two new approaches for estimating local lower bounds of specific entropy, which prove effective for both smooth and discontinuous problems. Numerical experiments demonstrate that our entropy-preserving methods maintain high-order accuracy while effectively suppressing spurious oscillations, outperforming existing local entropy minimum estimation techniques in the literature. Moreover, our approach is not limited to the relativistic Euler equations but can also be applied to other hydrodynamic models that admit an MEP.