修正模板改进的五阶WENO-Z+格式

doi:10.19596/j.cnki.1001-246x.8654

摘要/Abstract

摘要：

介绍一种修正模板近似方法, 即改进经典五阶WENO-JS格式中各候选子模板上数值通量的二阶多项式逼近, 通过加入三次修正项使模板逼近达到四阶精度, 并且通过引入可调函数使其具有ENO性质。将这种修正模板近似方法运用到WENO-Z+和WENO-Z+M格式中, 发展基于修正模板类型的WENO-Z+格式(WENO-MS-Z+, WENO-MS-Z+M)。通过数值算例对格式进行数值测试, 结果显示: 该格式具有强健的激波捕捉能力和对小尺度波结构的高分辨率, 与原WENO-Z+和WENO-Z+M格式相比改进明显。

关键词: 双曲守恒律, WENO-Z+, 修正模板, 非线性权

Abstract:

First, a modified stencil approximation method is introduced, which improves the second-order polynomial approximation of the numerical flux on each candidate sub-stencil in the classical fifth-order WENO-JS scheme. The stencil approximation reaches the fourth-order accuracy by adding a cubic correction term, and it has ENO property by introducing an adjustable function.Then the modified stencil approximation method is applied to WENO-Z+and WENO-Z+M schemes, and the modified WENO-Z+schemes based on the modified stencil type (WENO-MS-Z+, WENO-MS-Z+M) are developed.A series of numerical examples are used to test the new schemes. The results show that the new schemes have a strong ability to capture shock waves and high resolution for small-scale wave structures, which is significantly improved compared with the original WENO-Z+and WENO-Z+M schemes.

Key words: hyperbolic conservation laws, WENO-Z+, modified stencil, nonlinear weights

中图分类号:

O469

郭城, 程鹏丹, 王亚辉. 修正模板改进的五阶WENO-Z+格式[J]. 计算物理, 2023, 40(6): 666-676.

Cheng GUO, Pengdan CHENG, Yahui WANG. Improved Fifth-order WENO-Z+ Schemes Based on Modified Stencil[J]. Chinese Journal of Computational Physics, 2023, 40(6): 666-676.

图/表 9

图1 五阶WENO数值通量模板

Fig.1 Stencils for the fifth-order WENO numerical flux

表1 线性对流方程(20)在式(21)初值①下，不同格式在t=2时的L1误差与收敛阶

Table 1 L1 errors and convergence rates at t=2 of different schemes for the linear advection Eq.(20) with the initial data ① of Eq.(21)

N	WENO-Z+		WENO-Z+M		WENO-MS-Z+		WENO-MS-Z+M
N	L¹ error	order	L¹ error	order	L¹ error	order	L¹ error	order
20	2.57 × 10^-4		3.40 × 10^-4		3.32 × 10^-4		3.13 × 10^-4
40	8.04 × 10^-6	5.00	9.60 × 10^-6	5.15	9.20 × 10^-6	5.17	9.01 × 10^-6	5.12
80	2.44 × 10^-7	5.04	2.56 × 10^-7	5.23	2.50 × 10^-7	5.20	2.44 × 10^-7	5.21
160	8.07 × 10^-9	4.92	7.61 × 10^-9	5.07	7.21 × 10^-9	5.11	6.97 × 10^-9	5.13
320	2.47 × 10^-10	5.03	2.38 × 10^-10	5.00	2.16 × 10^-10	5.06	2.06 × 10^-10	5.08

表2 线性对流方程(20)在式(21)初值②下，不同格式在t=2时的L1误差与收敛阶

Table 2 L1 errors and convergence rates at t=2 of different schemes for the linear advection Eq.(20) with the initial data ② of Eq.(21)

N	WENO-Z+		WENO-Z+M		WENO-MS-Z+		WENO-MS-Z+M
N	L¹ error	order	L¹ error	order	L¹ error	order	L¹ error	order
20	4.36 × 10^-3		4.19 × 10^-3		1.95 × 10^-3		1.75 × 10^-3
40	1.37 × 10^-4	4.98	1.31 × 10^-4	4.99	6.10 × 10^-5	4.99	5.47 × 10^-5	5.00
80	4.58 × 10^-6	5.00	3.58 × 10^-6	5.19	1.56 × 10^-6	5.29	1.50 × 10^-6	5.19
160	1.43 × 10^-7	5.00	1.16 × 10^-7	4.95	4.41 × 10^-8	5.14	4.23 × 10^-8	5.15
320	4.44 × 10^-9	5.03	3.57 × 10^-9	5.02	1.38 × 10^-9	5.00	1.32 × 10^-9	5.00

图2 Sod激波管问题的数值密度结果(t=2, N=200, CFL=0.6)

Fig.2 Density contours of Sod shock tube problem on 200 grid points at t=2, CFL=0.6

图3 Lax激波管问题的数值密度结果(t=1.3, N=200, CFL=0.6)

Fig.3 Density contours of Lax shock tube problem on 200 grid points at t=1.3, CFL=0.6

图4 冲击波的相互作用的密度等值线图(t=0.038, N=400, CFL=0.6)

Fig.4 Density contours of Shock wave interaction problem on 400 grid points at t=0.038, CFL=0.6

图5 激波等熵波相互作用(Shu-Osher)的密度分布(t=1.8, CFL=0.6, N=400)

Fig.5 Density contours of Shock isentropic wave interaction (shu-osher) problem at t=1.8, N=400, CFL=0.6

图6 2D Riemann问题的密度等值线(Δx=Δy=1/400, CFL=0.5, t=0.8) (a)WENO-Z+; (b)WENO-MS-Z+; (c)WENO-Z+M; (d)WENO-MS-Z+M

Fig.6 Density contours of 2D Riemann problem on 401 × 401 grid points at t=0.8, CFL=0.5 (a) WENO-Z+; (b) WENO-MS-Z+; (c) WENO-Z+M; (d) WENO-MS-Z+M

图7 双马赫反射问题在t=0.2的密度等值线(网格点为961 × 241, CFL=0.5) (a)WENO-Z+; (b)WENO-MS-Z+; (c)WENO-Z+M; (d)WENO-MS-Z+M

Fig.7 Density contours of double Mach reflection problem on 961 × 241 grid points at t=0.2, CFL=0.5 (a) WENO-Z+; (b) WENO-MS-Z+; (c) WENO-Z+M; (d) WENO-MS-Z+M

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