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High-order well-balanced positivity-preserving affine-invariant finite volume CWENOZ scheme with LDCU flux for the Euler equations with gravitation.

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Title: High-order well-balanced positivity-preserving affine-invariant finite volume CWENOZ scheme with LDCU flux for the Euler equations with gravitation.
Authors: Xiang, Xiao-Shuo1 (AUTHOR) xiangxiaoshuo@stu.ouc.edu.cn, Li, Peng2 (AUTHOR) pengli@stdu.edu.cn, Gao, Zhen3 (AUTHOR) zhengao@ouc.edu.cn, Wang, Bao-Shan1,3 (AUTHOR) wbs@ouc.edu.cn
Source: Applied Numerical Mathematics. Jun2026, Vol. 224, p181-206. 26p.
Subjects: Finite volume method, Euler equations, Numerical analysis
Abstract: In this study, we propose a high-order well-balanced and positivity-reserving finite volume affine-invariant central weighted essentially non-oscillatory (Ai-CWENO) method for the Euler equations with gravitation. Unlike the dimension-by-dimension approach used in our previous work [Xiang et al., J. Sci. Comput., 2025, 102(2): 29], an affine-invariant CWENOZ operator based on fully two-dimensional basis functions is designed to reconstruct the two-dimensional polynomial directly. On the other hand, to reduce the dissipation, we adapt the newly developed low-dissipation central-upwind (LDCU) numerical flux in [Cui et al., J. Comput. Phys., 2025, 538: 114189] and rigorously prove its contact property, which is usually destroyed when using an incomplete Riemann solver. Then, the well-balanced property is achieved by using hydrostatic reconstruction and approximation of the source term. For extreme problems, we further incorporate the positivity-preserving limiter to ensure the stability of numerical solutions and theoretically analyze the positivity-preserving properties with a suitable CFL condition. Several numerical examples demonstrate that the proposed scheme achieves fifth-order accuracy for smooth problems, maintains the well-balanced property at the steady state, accurately captures small perturbations near the steady-state solution, and preserves the positivity of the density and pressure. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:In this study, we propose a high-order well-balanced and positivity-reserving finite volume affine-invariant central weighted essentially non-oscillatory (Ai-CWENO) method for the Euler equations with gravitation. Unlike the dimension-by-dimension approach used in our previous work [Xiang et al., J. Sci. Comput., 2025, 102(2): 29], an affine-invariant CWENOZ operator based on fully two-dimensional basis functions is designed to reconstruct the two-dimensional polynomial directly. On the other hand, to reduce the dissipation, we adapt the newly developed low-dissipation central-upwind (LDCU) numerical flux in [Cui et al., J. Comput. Phys., 2025, 538: 114189] and rigorously prove its contact property, which is usually destroyed when using an incomplete Riemann solver. Then, the well-balanced property is achieved by using hydrostatic reconstruction and approximation of the source term. For extreme problems, we further incorporate the positivity-preserving limiter to ensure the stability of numerical solutions and theoretically analyze the positivity-preserving properties with a suitable CFL condition. Several numerical examples demonstrate that the proposed scheme achieves fifth-order accuracy for smooth problems, maintains the well-balanced property at the steady state, accurately captures small perturbations near the steady-state solution, and preserves the positivity of the density and pressure. [ABSTRACT FROM AUTHOR]
ISSN:01689274
DOI:10.1016/j.apnum.2026.02.001