High-Order Accurate Structure-Preserving Finite Volume Scheme for Ten-Moment Gaussian Closure Equations with Source Terms: Positivity and Well-Balancedness

Abstract This paper develops high-order accurate positivity-preserving (PP) and well-balanced (WB) finite volume schemes for the ten-moment Gaussian closure equations with source terms based on the prior knowledge of the equilibrium states, extending the work on the discontinuous Galerkin method in [J. Wang, H. Tang and K. Wu, High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms, J. Comput. Phys. 519 2024, Article ID 113451]. The semi-discrete schemes are constructed based on the Harten-Lax-van Leer-Contact (HLLC) flux with modified solution states, along with suitable discretization and decomposition of the source terms. The fully-discrete schemes obtained by the explicit strong-stability-preserving Rung-Kutta time discretizations (including the forward Euler) can be proved to maintain the WB property for a given known hydrostatic equilibrium state. A rigorous PP analysis for the fully-discrete schemes is provided, based on several key properties of the HLLC flux and the admissible state set as well as the geometric quasilinearlization (GQL) approach [K. Wu and C.-W. Shu, Geometric quasilinearization framework for analysis and design of bound-preserving schemes, SIAM Rev. 65 2023, 4, 1031–1073] and [K. Wu and H. Tang, Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations, Math. Models Methods Appl. Sci. 27 2017, 10, 1871–1928] transforming complex nonlinear constraints in the admissible state set into simple linear ones. Based on several newly introduced properties of the HLLC flux, we may not decompose the high-order schemes into a convex combination of the “first-order schemes”, which permits us to skip the proof of the PP property for the first-order scheme and directly analyze the high-order schemes. Consequently, the present PP analysis is more simple and direct compared to that in Wang, Tang and Wu (2024). Several numerical experiments validate the high-order accuracy, WB and PP properties as well as the high resolution of the proposed schemes.