Ripa方程的高阶平衡有限差分A-WENO格式

High-Order Well-Balanced Finite Difference A-WENO Scheme for the Ripa Equations

摘要: 在本文中，我们提出了一种基于自适应加权本质无振荡(Adaptive Weighted Essentially Non-Oscillatory, A-WENO)方法的高阶有限差分格式，用于求解在浅水方程中引入水平温度梯度后形成的Ripa方程。该方程属于带源项的非齐次双曲守恒律，其稳态解在通量梯度与源项之间存在严格平衡。所提出的数值格式在有限差分框架下，通过对源项进行特殊分裂、结合改进的Lax-Friedrichs通量分裂及特征分解，并将由WENO重建得到的离散导数系数“冻结”后应用于源项离散，从而在保持五阶空间精度的同时，实现了严格的稳态保持性质。与传统的高阶WENO格式相比，本文方法在复杂底形和强非线性条件下能够有效抑制数值振荡，并显著提高了计算的稳定性与精度。数值算例验证了该方法在保持平衡、高阶精度及捕捉间断方面的优越性能。

Abstract: In this paper, a high-order finite difference scheme based on the Adaptive Weighted Essentially Non-Oscillatory (A-WENO) method is proposed for solving the Ripa equations, which are derived from the shallow water equations by introducing horizontal temperature gradients. The Ripa system represents a class of nonhomogeneous hyperbolic conservation laws with source terms, whose steady-state solutions exhibit an exact balance between the flux gradients and the source terms. Within the finite difference framework, the proposed method reformulates the source term, incorporates a modified Lax–Friedrichs flux splitting and characteristic decomposition, and applies the frozen discrete derivative coefficients obtained from WENO reconstruction to the source term discretization. This approach achieves strict well-balanced properties while maintaining fifth-order spatial accuracy. Compared with conventional high-order WENO schemes, the proposed A-WENO method effectively suppresses numerical oscillations under complex topographies and strong nonlinear conditions, enhancing both stability and accuracy. Numerical experiments demonstrate that the method exhibits excellent well-balanced behavior, high-order accuracy, and strong capability in resolving discontinuities.