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 引用本文: 武从海, 李虎, 刘旭亮, 罗勇, 张树海. 7阶WENO-S格式的计算效率研究. 力学学报, 2023, 55(1): 239-253.
Wu Conghai, Li Hu, Liu Xuliang, Luo Yong, Zhang Shuhai. Investigation of the time efficiency of the seventh-order WENO-S scheme. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 239-253.
 Citation: Wu Conghai, Li Hu, Liu Xuliang, Luo Yong, Zhang Shuhai. Investigation of the time efficiency of the seventh-order WENO-S scheme. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 239-253.

## INVESTIGATION OF THE TIME EFFICIENCY OF THE SEVENTH-ORDER WENO-S SCHEME

• 摘要: WENO-S格式是一类适合于含间断问题数值模拟的加权本质无振荡格式. 这类格式的光滑因子满足对单频波为常数, 这使得其近似色散关系与线性基底格式一致, 并且具有良好的小尺度波动模拟能力．计算效率是数值方法性能指标的一个重要方面. 由于WENO-S格式的光滑因子在各子模板上的计算公式除下标不同外形式一致, 在计算线性对流方程相邻数值通量时, 部分光滑因子完全相同．为此提出一种消除WENO-S格式冗余光滑因子计算的方法. 该方法要求一条网格线上用于重构或插值的量可以表示为一个序列. 基于此要求分析其对于几种不同物理问题的可行性和使用方法. 以7阶WENO-S格式为例介绍了格式性质和去除冗余光滑因子计算的方法. 该方法中预先计算和存储一条网格线上的所有光滑因子, 在网格点较多的情况下, 光滑因子计算次数约为原7阶WENO-S格式的1/4. 对一维对流问题、球面波传播问题、二维旋转问题、二维小扰动传播问题及一维和二维无黏流动问题进行了数值模拟. 结果表明该格式对多种流动结构具有良好的捕捉能力, 并且同时具有良好的计算效率, 去除冗余计算后又降低了约20%的计算时间.

Abstract: The WENO-S scheme is a class of weighted essentially non-oscillatory schemes suitable for numerical simulations of problems with discontinuities. The smoothness indicator of this kind of scheme is constant for single-frequency waves, which makes this kind of scheme have exactly the same approximate dispersion relationship with its linear base scheme, and thus has an excellent ability to simulate small-scale waves. Time efficiency is crucial for numerical methods. For a WENO-S scheme, the formula of the smoothness indicator on each sub-stencil has the same formula except for different subscripts. Then some smoothness indicators are the same when calculating adjacent numerical fluxes of linear convection equations. So, a method is proposed to remove redundant computations of smoothness indicators. The premise of this approach is that the quantity used for reconstruction or interpolation on a grid line can be represented as a sequence. According to this requirement, the feasibility and application requirements for several different physical problems are analyzed. The seventh-order WENO-S scheme is employed to illustrate the advantages of the WENO-S schemes, including good properties near extreme points, good stability near discontinuities, and outstanding spectral properties. Then the method of eliminating the computation of the redundant smoothness indicators is introduced. In numerical computation, all smoothness indicators in a grid line are calculated and stored in advance. With this approach, the count of the smoothness indicator calculation is about 1/4 of the original one for the seventh-order WENO-S scheme when there are many grid points. Numerical examples include one-dimensional advection, spherical wave propagation, two-dimensional rotation, small disturbance propagation, and one- and two- dimensional inviscid flow problems. The numerical results show that this scheme can capture a variety of flow structures well and have good time efficiency. Furthermore, the proposed method reduces the computational time by about 20%.

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