改进的五阶WENO-Z-格式

骆信; 吴颂平

doi:10.6052/0459-1879-19-249

改进的五阶WENO-Z-格式

doi: 10.6052/0459-1879-19-249

骆信,
吴颂平

北京航空航天大学航空科学与工程学院, 北京 100191

基金项目: 1) 国家自然科学基金资助项目(91530325)

详细信息

通讯作者:
吴颂平

中图分类号: V211.3
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出版历程
- 收稿日期: 2019-09-04
- 刊出日期: 2019-11-18

AN IMPROVED FIFTH-ORDER WENO-Z+ SCHEME

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

摘要

摘要: WENO-Z$+\!$格式的性能提升依赖于新增项的作用,该项的作用是在WENO-Z格式的基础上进一步增大欠光滑子模板上的权重. 系数$\lambda$被设置用来调控该项的作用, 以避免负耗散. 本文指出了WENO-Z$+\!$格式的缺陷,其所采用$\lambda $的取值方式既不能充分发挥格式的潜力, 也未完全消除负耗散;提出$\lambda $的值应随当地流场数据变化,方能充分发挥新增项在降低数值耗散、提高分辨率上的潜力. 基于此,本文重新设计了$\lambda $的计算公式,该公式能自适应地调控新增项的作用: 只在欠光滑子模板上的权重容易过度增大的地方削弱该项的作用,以避免负耗散; 在其他地方则充分发挥新增项的作用,最大限度增大欠光滑子模板上的权重, 提高格式的分辨率.将使用该系数公式的新格式命名为WENO-Z++, 并对其数值性能进行了系统的研究.理论分析表明, 新格式在间断处具有基本无振荡(essentially non-oscillatory,ENO)特性和更低的数值耗散. 对近似色散关系(approximate dispersion relation,ADR)的研究表明,新格式有效地避免了因过度增大欠光滑子模板上的权重而带来的负耗散,其频谱特性也得到了显著提升.本文还推导了使新格式在极值点处也能保持最优阶的精度的参数设置.一系列求解Euler方程的数值试验表明,新格式的激波捕捉能力和对复杂流场结构的分辨率都显著优于原WENO-Z$+\!$格式.}
- WENO格式 /
- 频谱特性 /
- 高分辨率 /
- 数值耗散 /
- 激波捕捉 /
- 复杂流动
Abstract: The performance improvement of the WENO-Z+ scheme depends on the role of the additional term, which is added to the WENO-Z weights to increase the weights of less-smooth substencils further. Since the additional term may lead to negative dissipation by over increasing the weights of less-smooth substencils in smooth regions, the coefficient $\lambda $ is set to control the role of this term and needs to be carefully determined. In this paper, the defects of the method the WENO-Z+ scheme adopts to determine the value of $\lambda $ are pointed out: It can neither fully utilize the potential of the scheme nor effectively avoid negative dissipation. We propose that to take the full role of the additional term in reducing numerical dissipation and improving resolution ability, the value of $\lambda $ should change with the local data of the flow field. Based on this idea, we design a new calculation formula for $\lambda $, which can adjust the role of the additional term adaptively: Weaken the role of the additional term only where the weights of less-smooth substencils are easy to be excessively increased. The new scheme employing the new $\lambda $ formula is named WENO-Z++, and its numerical performance is systematically analyzed. Theoretical analysis indicates that the new scheme maintains essentially non-oscillatory (ENO) property and has lower numerical dissipation at discontinuities. The investigation of approximate dispersion relation (ADR) shows that the new scheme effectively avoids the negative dissipation caused by excessive increase of the weights of less-smoothed substencils, and its spectral properties are significantly improved. The parameters set that allow the new scheme keeping the optimal order of accuracy at extreme points is deduced. A series of numerical experiments for solving the Euler equations show that both the shock-capturing ability and resolution for complex flow structure of the new scheme are significantly better than those of the original WENO-Z+ scheme.
- WENO scheme /
- spectral property /
- high resolution /
- numerical dissipation /
- shock capturing /
- complex flow

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