浅水方程组的高精度WENO格式及其应用

High-accuracy WENO scheme for shallow water equations and its applications

  • 摘要: 本文通过对经典WENO-JS格式中三个小模版光滑因子进行分解,将分解后的二阶导数项重新组合,得到精度更高的全局光滑因子,以此建立一个新的、低耗散的五阶精度WENO格式。首先,通过泰勒展开式证明该格式在一阶和二阶极值点处仍保持五阶精度;其次,一维对流方程的连续初始条件验证了格式的高精度特性,一维对流方程的间断初始条件验证了格式的高分辨率特性,一维欧拉方程组激波管问题验证了格式对非线性方程组计算的数值稳定性;最后,基于浅水方程组建立一、二维溃坝洪水模型,利用本文格式对经典算例溃坝水流进行数值模拟,并与其他格式模拟结果对比,结果表明新格式对强间断问题的模拟效果明显优于同阶其他格式。因此,本文所建立的高精度WENO格式是一个性能优良的激波捕捉格式,可用于其他间断问题的数值模拟。

     

    Abstract: In this paper, we propose a novel low-dissipation fifth-order WENO (Weighted Essentially Non-Oscillatory) scheme by decomposing the three small modal smoothness indicators in the classical WENO-JS scheme and recombining the resulting second-order derivative terms to construct a more accurate global smoothness indicator. First, we demonstrate through Taylor series expansion that the proposed scheme preserves fifth-order accuracy at both first-order and second-order critical points. Next, we verify its high-accuracy properties using continuous initial conditions for one-dimensional convection equations, and assess its high-resolution capability under discontinuous initial conditions for the same equations. The numerical stability of the scheme for nonlinear systems is further validated through the one-dimensional Euler equations shock-tube problem. Finally, we establish one- and two-dimensional dam-break flood models based on the shallow water equations and simulate the classical dam-break flow using the proposed scheme. Comparative results with other schemes show that our method significantly outperforms existing approaches in capturing strong intermittent phenomena. Therefore, the high-accuracy WENO scheme presented in this study is a high-performance tool for surge modeling and can be effectively applied to the numerical simulation of other intermittent flow problems.

     

/

返回文章
返回