二维扩散方程的局部二阶线性保极值节点计算方法及应用

doi:10.19596/j.cnki.1001-246x.8337

摘要/Abstract

摘要：

对一般四边形网格设计一种优化的节点控制体, 并构造了一种扩散方程的保极值二阶收敛的局部线性节点计算格式(优化控制体节点格式, VOC格式)。在网格不出现异常节点的情况下, 证明VOC格式是保极值、线性精确和二阶收敛的。而且在均匀的矩形网格上, 修正的逆距离加权格式与VOC格式等价, 从而对间断系数问题也是局部二阶收敛的。VOC格式可以用于单元中心型线性扩散格式和保正格式的节点值计算。数值算例表明对扭曲网格上的间断系数问题, VOC格式是二阶收敛的。采用VOC格式计算节点值的线性九点格式具有线性精确性和二阶收敛性, 采用VOC格式的保正格式也具有二阶收敛性。

关键词: 节点格式, 修正的逆距离加权格式, 有限体积方法, 扩散方程, 保极值

Abstract:

We construct a kind of optimized control volume for general quadrilateral meshes. Based on the control volume, we devise a local linear vertex scheme (Vertex-scheme on Optimized Control volume, VOC), which is maximum-principle-preserving, and 2nd order convergent. We prove that VOC scheme is maximum-principle-preserving, linearity-preserving and 2nd order convergent if no exceptional vertex exists. On uniform rectangle meshes, we prove that the modified inverse distance weight (MIDW) scheme is approximately VOC scheme, and they are all 2nd order convergent. VOC scheme can be used to construct linear cell-centered diffusion schemes and positivity-preserving diffusion schemes. Numerical experiments verify that the scheme is 2nd order convergent on distorted meshes for diffusion equations with discontinuous coefficients. The linear scheme which adopts VOC scheme is linearity-preserving and 2nd order convergent, and the positivity-preserving scheme is also 2nd order convergent.

Key words: vertex schemes, modified inverse distance weight, finite volume method, diffusion equation, maximum-principle-preserving

张淑娴, 杭旭登. 二维扩散方程的局部二阶线性保极值节点计算方法及应用[J]. 计算物理, 2022, 39(1): 17-32.

Shuxian ZHANG, Xudeng HANG. Applications of Local Maximum-principle-preserving Linear Vertex Scheme of Second Order Convergence for Two-dimensional Diffusion Equations[J]. Chinese Journal of Computational Physics, 2022, 39(1): 17-32.

图/表 27

参考文献 22

1	HANG X D , LI J H , YUAN G W . Assistant mesh difference method for parabolic equations on distorted meshes with large aspect ratio[J]. Chinese J Comput Phys, 2007, 24 (3): 268- 276.
2	YUAN G W , HANG X D . Tangential flux on grid edge in nine-point schemes for diffusion equations[J]. Chinese J Comput Phys, 2008, 25 (1): 7- 14.
3	ZHAO F , SHENG Z Q , YUAN G W . A positivity-preserving finite volume scheme based on second-order scheme[J]. Chinese J Comput Phys, 2020, 37 (4): 379- 392.
4	LI D Y , SHUI H S , TANG M J . On the finite difference scheme of two-dimensional parabolic equation in a non-rectangular mesh[J]. J Numer Methods Comput Appl, 1980, 1 (4): 217- 224.
5	SHENG Z Q , YUAN G W . A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes[J]. SIAM J Sci Comput, 2008, 30 (3): 1341- 1361. DOI
6	YUAN G W , SHENG Z Q . Monotone finite volume schemes for diffusion equations on polygonal meshes[J]. J Comput Phys, 2008, 227 (12): 6288- 6312. DOI
7	BASKO M M , MARUHN J , TAUSCHWITZ A . An efficient cell-centered diffusion scheme for quadrilateral grids[J]. J Comput Phys, 2009, 228 (6): 2175- 2193. DOI
8	BERTOLAZZ E , MANZINI G . On vertex constructions for cell-centered finite volume approximations of 2D anisotropic diffusion problems[J]. Math Model Meth Appl Sci, 2007, 17 (1): 1- 32. DOI
9	COUDIÈRE Y , VILA J-P , VILLEDIEU P . Convergence rate of a finite volume scheme for a two-dimensional diffusion convection problem[J]. Math Model Numer Anal, 1999, 33 (3): 493- 516. DOI
10	HAN C H , KANG T , WU J M . A linearity preserving difference scheme for the numerical solution of diffusion equation on distorted meshes[J]. J Commun Univ China (Sci Tech), 2008, 15 (1): 40- 44.
11	WU J M , DAI Z H , GAO Z M , et al. Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes[J]. J Comput Phys, 2010, 229 (9): 3382- 3401. DOI
12	WU J M , GAO Z M . A nine-point scheme with explicit weights for diffusion equations on distorted meshes[J]. Appl Numer Math, 2011, 61 (7): 844- 867. DOI
13	ZHAO Q , YUAN G W . Analysis and construction of cell-centered finite volume scheme for diffusion equations on distorted meshes[J]. Comput Methods Appl Mech Engrg, 2009, 198 (37-40): 3039- 3050. DOI
14	SHENG Z Q , YUAN G W . A finite volume scheme for diffusion equations on distorted quadrilateral meshes[J]. Transport Theory and Statistical Physics, 2008, 37 (2-4): 171- 207. DOI
15	NORDBOTTEN J M , AAVATSMARK I , EIGESTAD G T . Monotonicity of control volume methods[J]. Numer Math, 2007, 106 (2): 255- 288. DOI
16	SHENG Z Q , YUAN G W . A new nonlinear finite volume scheme preserving positivity for diffusion equations[J]. J Comput Phys, 2016, 315: 182- 193. DOI
17	SU S , DONG Q N , WU J M . A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes[J]. J Comput Phys, 2018, 372: 773- 798. DOI
18	BLANC X , LABOURASSE E . A positive scheme for diffusion problems on deformed meshes[J]. Z Angew Math Mech, 2016, 96 (6): 660- 680. DOI
19	CHAI G L , SU J W , WANG L . An inverse distance weighting spatial interpolation algorithm with second order accuracy[J]. Chinese J Comput Phys, 2020, 37 (4): 393- 402.
20	AGELAS L , EYMARD R , HERBIN R . A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media[J]. C R Acad Sci Paris Ser, 2009, 347 (11/12): 673- 676.
21	CHANG L N , YUAN G W . An efficient and accurate reconstruction algorithm for the formulation of cell-centered diffusion schemes[J]. J Comput Phys, 2012, 231 (20): 6935- 6952. DOI
22	EYMARD R, GALLOIIET T, HERBIN R. Finite volume methods[M]//CIARLET P G, LIONS J L, eds. Handbook of numerical analysis Vol. Ⅶ, North-Holland, 2000.

M × N		12 × 12	24 × 24	48 × 48	96 × 96	192 × 192
MIDW	L^∞	3.19×10^-2	1.59×10^-2	9.35×10^-3	5.03×10^-3	2.83×10^-3
	Rate		1.01	0.76	0.89	0.83
	L²	9.99×10^-3	5.21×10^-3	2.56×10^-3	1.32×10^-3	6.71×10^-4
	Rate		0.94	1.02	0.96	0.98
Taylor	L^∞	4.12×10^-3	1.15×10^-3	3.15×10^-4	7.87×10^-5	2.02×10^-5
	Rate		1.84	1.87	2.00	1.96
	L²	2.95×10^-3	7.75×10^-4	2.00×10^-4	5.05×10^-5	1.27×10^-5
	Rate		1.93	1.95	1.99	1.99
VOC-N	L^∞	4.12×10^-3	1.13×10^-3	2.95×10^-4	7.83×10^-5	1.98×10^-5
	Rate		1.87	1.94	1.91	1.98
	L²	2.93×10^-3	7.65×10^-4	1.97×10^-4	4.98×10^-5	1.25×10^-5
	Rate		1.94	1.95	1.99	1.99
VOC-Y	L^∞	3.33×10^-3	8.84×10^-4	2.34×10^-4	6.30×10^-5	1.62×10^-5
	Rate		1.91	1.92	1.89	1.96
	L²	2.32×10^-3	6.09×10^-4	1.56×10^-4	3.96×10^-5	9.94×10^-6
	Rate		1.93	1.96	1.98	1.99

M × N		12 × 12	24 × 24	48 × 48	96 × 96	192 × 192
MIDW	L^∞	3.19×10^-2	1.59×10^-2	9.35×10^-3	5.03×10^-3	2.83×10^-3
	Rate		1.01	0.76	0.89	0.83
	L²	9.99×10^-3	5.21×10^-3	2.56×10^-3	1.32×10^-3	6.71×10^-4
	Rate		0.94	1.02	0.96	0.98
Taylor	L^∞	4.12×10^-3	1.15×10^-3	3.15×10^-4	7.87×10^-5	2.02×10^-5
	Rate		1.84	1.87	2.00	1.96
	L²	2.95×10^-3	7.75×10^-4	2.00×10^-4	5.05×10^-5	1.27×10^-5
	Rate		1.93	1.95	1.99	1.99
VOC-N	L^∞	4.12×10^-3	1.13×10^-3	2.95×10^-4	7.83×10^-5	1.98×10^-5
	Rate		1.87	1.94	1.91	1.98
	L²	2.93×10^-3	7.65×10^-4	1.97×10^-4	4.98×10^-5	1.25×10^-5
	Rate		1.94	1.95	1.99	1.99
VOC-Y	L^∞	3.33×10^-3	8.84×10^-4	2.34×10^-4	6.30×10^-5	1.62×10^-5
	Rate		1.91	1.92	1.89	1.96
	L²	2.32×10^-3	6.09×10^-4	1.56×10^-4	3.96×10^-5	9.94×10^-6
	Rate		1.93	1.96	1.98	1.99

M × N		12 × 12	24 × 24	48 × 48	96 × 96	192 × 192
MIDW	L^∞	1.56	1.04	5.42	2.36	1.37
	Rate		0.58	0.94	1.20	0.78
	L²	3.85	2.17	1.14	5.68×10^-1	2.89×10^-1
	Rate		0.83	0.93	1.00	0.98
Taylor	L^∞	2.11×10¹	1.23×10¹	6.09	3.21	1.60
	Rate		0.78	1.01	0.93	1.00
	L²	5.10	2.20	7.58×10^-1	2.73×10^-1	9.44×10^-2
	Rate		1.21	1.54	1.47	1.53
VOC-N	L^∞	2.13×10^-2	6.58×10^-3	1.95×10^-3	4.99×10^-4	1.48×10^-4
	Rate		1.70	1.75	1.97	1.75
	L²	6.62×10^-3	1.83×10^-3	4.75×10^-4	1.21×10^-4	3.05×10^-5
	Rate		1.86	1.94	1.98	1.98
VOC-Y	L^∞	1.71×10^-2	5.19×10^-3	1.72×10^-3	4.25×10^-4	1.33×10^-4
	Rate		1.72	1.59	2.02	1.67
	L²	5.17×10^-3	1.44×10^-3	3.78×10^-4	9.61×10^-5	2.45×10^-5
	Rate		1.84	1.93	1.97	1.97

M × N		12 × 12	24 × 24	48 × 48	96 × 96	192 × 192
MIDW	L^∞	1.56	1.04	5.42	2.36	1.37
	Rate		0.58	0.94	1.20	0.78
	L²	3.85	2.17	1.14	5.68×10^-1	2.89×10^-1
	Rate		0.83	0.93	1.00	0.98
Taylor	L^∞	2.11×10¹	1.23×10¹	6.09	3.21	1.60
	Rate		0.78	1.01	0.93	1.00
	L²	5.10	2.20	7.58×10^-1	2.73×10^-1	9.44×10^-2
	Rate		1.21	1.54	1.47	1.53
VOC-N	L^∞	2.13×10^-2	6.58×10^-3	1.95×10^-3	4.99×10^-4	1.48×10^-4
	Rate		1.70	1.75	1.97	1.75
	L²	6.62×10^-3	1.83×10^-3	4.75×10^-4	1.21×10^-4	3.05×10^-5
	Rate		1.86	1.94	1.98	1.98
VOC-Y	L^∞	1.71×10^-2	5.19×10^-3	1.72×10^-3	4.25×10^-4	1.33×10^-4
	Rate		1.72	1.59	2.02	1.67
	L²	5.17×10^-3	1.44×10^-3	3.78×10^-4	9.61×10^-5	2.45×10^-5
	Rate		1.84	1.93	1.97	1.97

M × N		12 × 12	24 × 24	48 × 48	96 × 96	192 × 192
MIDW	L^∞	1.12×10^-1	5.76×10^-2	3.31×10^-2	3.48×10^-2	3.54×10^-2
	Rate		0.96	0.80	-0.07	-0.02
	L²	2.77×10^-2	1.37×10^-2	1.50×10^-2	1.69×10^-2	1.73×10^-2
	Rate		1.02	-0.14	-0.17	-0.03
Taylor	L^∞	1.31×10^-1	3.96×10^-2	1.09×10^-2	3.61×10^-3	1.89×10^-3
	Rate		1.73	1.86	1.60	0.94
	L²	3.05×10^-2	7.62×10^-3	1.88×10^-3	4.93×10^-4	1.31×10^-4
	Rate		2.00	2.02	1.93	1.92
VOC-Y	L^∞	1.32×10^-1	3.80×10^-2	1.19×10^-2	2.54×10^-3	7.33×10^-4
	Rate		1.79	1.68	2.23	1.79
	L²	3.10×10^-2	7.81×10^-3	1.89×10^-3	4.61×10^-4	1.20×10^-4
	Rate		1.99	2.05	2.03	1.95