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Hybrid finite difference schemes without free parameters
WANG Lai, WU Songping
School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:A new algorithm was proposed to calculate the weights of sub-schemes in hybrid compact-WENO (weighted essentially non-oscillatory) schemes. It used smoothness indicators of the stencils of hybrid schemes as variables to work globally. The new weight algorithm employed none freely user-defined parameter, making the hybrid schemes more robust and applicable. Two different 5th order upwind compact schemes were coupled with a 5th order WENO-Z scheme respectively. The new algorithm maked the transition between two sub-schemes more smoothly. Generally, WENO-Z could preserve the accuracy at critical points while compact schemes were less dissipative in smooth area. Thus, these hybrid schemes with new algorithm maintained high resolution and high order of accuracy qualities and shock-capturing abilities. Numerical tests were taken upon benchmarks of hyperbolic equations to test performances of these two hybrid schemes between the new weight algorithm and a typical one. Numerical results demonstrate that the new parameter-free weight algorithm performs well.
Key words: hybrid scheme     no free parameters     weight algorithm     high resolution     low dissipation

1 混合格式的构造 1.1 控制方程

1.2 守恒型紧致格式

1.3 WENO-Z格式

1.4 两种子格式的混合

Pirozzoli在文献[11]中定义了一个简单的权重算子,这种算子造成了混合格式中两种子格式的切换过于突然,切换点会产生不容忽视的数值振荡乃至污染流场.为了克服上述问题Ren[12]和武从海等[14]提出了一个新的权重算子：

2 数值试验结果

2.1 激波管问题

LAX问题是1D激波管问题的典型算例之一.在区间[-5.0,5.0]之内,以原点为分界点,左右两侧的气体初始状态不同.初始条件如下：

 图 1 HCW-U,HCW-UL,LAX问题,密度值Fig. 1 HCW-U,HCW-UL,LAX problem,density
 图 2 HCW-U,HCW-UL,LAX问题,压强值Fig. 2 HCW-U,HCW-UL,LAX problem,pressure
2.2 激波与熵波的干涉

 图 3 HCW-U,Osher-Shu问题,区间[5,7.4]密度值Fig. 3 HCW-U,Osher-Shu problem,density in[5,7.4]
 图 4 HCW-UL,Osher-Shu问题,区间[5,7.4]密度值Fig. 4 HCW-UL,Osher-Shu problem,density in[5,7.4]
2.3 2D双马赫反射

 图 5 HCW-U-R,DMR问题,密度云图,50条等值线Fig. 5 HCW-U-R,DMR problem,density contour,50 levels
 图 6 HCW-U-Z,DMR问题,密度云图,50条等值线Fig. 6 HCW-U-Z,DMR problem,density contour,50 levels
 图 7 HCW-UL-Z,DMR问题,密度云图,50条等值线Fig. 7 HCW-UL-Z,DMR problem,density contour,50 levels
3 结 论

1) 新权重算子(Z型)对间断的捕捉能力良好,同时相比R型权重能够抑制间断处数值振荡的传播.

2) 相比于WENO-Z,采用Z型权重算子的混合格式HCW-Z耗散降低,分辨率提高.HCW-U与HCW-UL的数值结果并无明显区别.

3) Z型权重算子的数值特性与R型权重算子(rc=0.5)相比区别不明显,但是R型权重算子的计算效果依赖rc的选取，使用受到局限.

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#### 文章信息

WANG Lai, WU Songping

Hybrid finite difference schemes without free parameters

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(2): 318-322.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0134