A modified entropy functional for improved high order shock capturing methods

This study proposes a new piecewise entropy functional aimed at improving entropy based troubled element detection in high order shock capturing methods under demanding near vacuum conditions. In such regimes, the classical thermodynamic entropy log ( p / ρ γ ) becomes singular and highly sensitive to small perturbations of the isentropic pressure, which can compromise the reliability of entropy thresholding strategies. The proposed functional is designed to reduce this stiffness while retaining the classical logarithmic entropy behaviour away from the low isentropic pressure regime. Building upon the generalized entropy framework introduced by Harten, the proposed functional is analysed in terms of compatibility and convexity. The analysis shows that it behaves as a valid convex entropy flux pair above a computable threshold in the isentropic pressure. Below this threshold, where strict convexity is lost, the functional is not interpreted as the basis for a global entropy stability proof, but rather as a regularized entropy indicator for troubled element detection. The functional is constructed to be piecewise continuously differentiable and is specifically tailored for use within modal filtering based shock capturing strategies for Discontinuous Galerkin methods. The only modification to the baseline filtering framework is the replacement of the original thermodynamic entropy used in the detection criterion with the proposed piecewise functional. Numerical experiments demonstrate that the modified entropy improves the reliability of troubled element detection in challenging near vacuum and high Mach number configurations, while preserving the robustness of the underlying shock capturing procedure.