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;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,并于其他格式模拟结果进行对比,结果表明本文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 achieve a more accurate global smoothness indicator. Firstly, we demonstrate through Taylor series expansion that the new scheme maintains fifth-order accuracy at both first-order and second-order critical points. Secondly, we validate the high-precision characteristics of the new scheme using continuous initial conditions for one-dimensional convective equations, and its high-resolution characteristics using discontinuous initial conditions for the same equations. Lastly, we establish one- and two-dimensional dam-break flood models based on the shallow water equations and simulate the classical dam-break flow using our new scheme. We compare these results with those obtained using other schemes, showing that our scheme performs significantly better in capturing strong intermittent phenomena. Therefore, the high-precision WENO scheme proposed in this paper is a high-performance format for capturing surges and can be effectively used for the numerical simulation of other intermittent problems.

     

/

返回文章
返回