This paper presents a high-order, finite volume, monotonicity-preserving WENO method that adheres to physical constraints (specifically, density \(\rho > 0\) and internal energy \(e > \frac{1}{2\rho ^2}\) ) for solving the Chaplygin gas dynamic equations. Due to the presence of negative pressure, the solutions may include delta waves, which pose significant numerical approximation challenges. Furthermore, negative pressure exacerbates the nonlinearity in the constraint on internal energy, leading to the invalidity of the typically expected Lax–Friedrichs (LF) splitting property. This complexity makes the design and analysis of our physical-constraint-preserving (PCP) schemes nontrivial. To address these challenges, we employ a novel geometric quasilinearization (GQL) approach recently proposed in [K. Wu & C.-W. Shu, SIAM Rev., 65(4):1031–1073, 2023]. Using GQL, we derive an equivalent linear representation of the admissible state set through the introduction of appropriate auxiliary variables. As alternatives to the invalid LF splitting property, we discover several critical inequalities and generalized LF splitting properties, thereby laying the groundwork for analyzing and designing PCP schemes. Thanks to these properties, PCP analysis is significantly simplified into seeking a summation of positive terms. To achieve high resolution for contact discontinuities and delta waves, we present a robust, high-order, monotonicity-preserving WENO method for the Chaplygin gas dynamics. The PCP property of this method is rigorously proven under conditions enforced by the PCP limiter. We conduct numerical experiments in one and two dimensions to evaluate the accuracy and robustness of our method. Our PCP method effectively captures the behavior of delta waves and contact discontinuities, even in scenarios with low density and internal energy.
