Applications of Local Maximum-principle-preserving Linear Vertex Scheme of Second Order Convergence for Two-dimensional Diffusion Equations

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

Abstract

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

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.

Figures/Tables 27

References 22

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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