保号WENO-AO型中心迎风格式

doi:10.19596/j.cnki.1001-246x.8507

摘要/Abstract

摘要：

证明Weighted Essentially Non-Oscillatory with Adaptive Order(WENO-AO)重构的保号性, 确保在单元交界面处重构值的跳跃符号与原始值的跳跃符号保持一致, 给出保号性成立的充分条件。WENO-AO重构通过高阶多项式与低阶重构的非线性组合来实现自适应收敛阶, 在求解不连续点附近的解时, WENO-AO重构比经典WENO重构更精确, 且具有很好的稳定性。采用中心迎风数值通量, 结合时间方向的三阶强稳定Runge-Kutta法计算。数值结果表明该格式最高可达五阶精度, 且具有分辨率高、鲁棒性强等良好特性, 并能够准确地捕捉间断位置, 有效抑制伪振荡的产生。

关键词: 保号性, WENO-AO重构, 双曲守恒律方程, 中心迎风通量函数

Abstract:

In this paper we give a sufficient condition for sign preservation in Weighted Essentially Non-Oscillatory with Adaptive Order (WENO-AO) reconstruction, which means to keep same sign for jumping at interfaces. WENO-AO reconstruction realizes self-adaptive accuracy through an nonlinear combination of high-order polynomial and low-order reconstruction. In solving solution near discontinuous points, WENO-AO reconstruction is more accurate than the classical WENO reconstruction, and it keeps good stability. The scheme is obtained by combining a central upwind numerical flux with a third-order strongly stable Runge-Kutta method. Numerical results show that the scheme has fifth order accuracy. It has characteristics of high resolution and strong robustness. It captures accurately discontinuous positions and avoids effectively the generation of spurious oscillation.

Key words: sign preserving, WENO-AO reconstruction, hyperbolic conservation laws equations, central upwind flux function

郑素佩, 建芒芒, 封建湖, 翟梦情. 保号WENO-AO型中心迎风格式[J]. 计算物理, 2022, 39(6): 677-686.

Supei ZHENG, Mangmang JIAN, Jianhu FENG, Mengqing ZHAI. Sign Preserving WENO-AO-type Central Upwind Schemes[J]. Chinese Journal of Computational Physics, 2022, 39(6): 677-686.

图/表 9

表1 光滑初始条件式(18)下线性对流方程的误差及收敛阶

Table 1 Error and convergence order of linear convection equation with smooth initial condition Eq.(18)

N	L₁ error	Rate	L_∞ error	Rate
40	2.237 9 × 10^-6		4.678 7 × 10^-6
80	6.025 7 × 10^-8	5.21	1.248 4 × 10^-7	5.22
160	1.730 5 × 10^-9	5.12	3.630 2 × 10^-9	5.10
320	5.168 8 × 10^-11	5.06	1.108 9 × 10^-10	5.03
640	1.578 3 × 10^-12	5.03	3.427 9 × 10^-12	5.01

表2 光滑初始条件式(19)下基于不同重构格式的误差及收敛阶

Table 2 Error and convergence order based on different reconstruction schemes under smooth initial condition Eq.(19)

N	WENO-AO		WENO5
N	L_∞ error	Rate	L_∞ error	Rate
20	2.680 1 × 10^-4		2.700 0 × 10^-3
40	6.382 7 × 10^-6	5.39	3.845 7 × 10^-4	2.81
80	1.703 4 × 10^-7	5.23	6.128 9 × 10^-5	2.65
160	4.952 5 × 10^-9	5.10	8.089 5 × 10^-6	2.92
320	1.514 0 × 10^-10	5.03	1.024 3 × 10^-6	2.98

图1 间断初始条件下线性对流方程的数值结果

Fig.1 Numerical results of linear convection equations with discontinuous initial conditions

表3 光滑初始条件下Burgers方程的误差及收敛阶

Table 3 Error and convergence order of Burgers equation with smooth initial conditions

N	L₁ error	Rate	L_∞ error	Rate
20	1.367 3 × 10^-4		1.839 4 × 10^-5
40	5.071 8 × 10^-6	4.75	6.035 5 × 10^-7	4.93
80	1.770 1 × 10^-7	4.84	1.886 4 × 10^-8	4.99
160	5.706 3 × 10^-9	4.96	5.900 7 × 10^-10	5.00
320	1.796 2 × 10^-10	4.99	1.845 1 × 10^-11	5.00

图2 间断初始条件下Burgers方程的数值结果

Fig.2 Numerical results of Burgers equation with discontinuous initial conditions

图3 溃坝问题的数值结果

Fig.3 Numerical results of dam break problem

图4 大型溃坝问题的数值结果

Fig.4 Numerical results of dam break problem

图5 Sod激波管问题的数值结果

Fig.5 Numerical results of Sod shock tube problem

图6 强稀疏波问题的数值结果

Fig.6 Numerical results of strong sparse wave problem

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