$ \theta $ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations

Yang Wang; Yi-fu Feng

doi:10.3934/naco.2019027

Article Contents

2019, Volume 9, Issue 4: 461-481. Doi: 10.3934/naco.2019027

This issue Previous Article System of generalized mixed nonlinear ordered variational inclusions Next Article A preconditioned SSOR iteration method for solving complex symmetric system of linear equations

$ \theta $ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations

Yang Wang^, and
Yi-fu Feng

School of Mathematics, Jilin Normal University, Siping 136000, P. R. China

^† Corresponding author
^† Corresponding author

Received: May 2018

Revised: March 2019

Published online: May 14, 2019

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Abstract

In this article, a $ \theta $ scheme based on wavelet-like incremental unknowns (WIU) is presented for a class of porous medium diffusion-type equations. Through some important norm inequalities, we prove the stability of $ \theta $ scheme. Compared to the classical scheme, the stability conditions are improved. Numerical results show that the $ \theta $ scheme based on the WIU decomposition is efficient.

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Mathematics Subject Classification: Primary: 65M06, 65P40; Secondary: 65M50.

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Figure 1. Coarse grid points(×)and fine grid points(○), d=1, N=4

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Figure 2. Coarse grid points(×), finer grid points(○) and the finest grid points (◇)

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Table 1. Comparison of CPU time and error with different $ d $ and $ N $ when $ \theta = 0 $

	$ M_1 $		$ M_2 $
	CPU	$ \\|{\rm{error}}\; \\| $	CPU	$ \\|{\rm{error}}\\|\; $
$ \tau=0.002,d=1,N=15 $	1.1719	1.2e-4	1.6094	4e-4
$ \tau=0.001,d=1,N=18 $	6.4844	1e-5	8.2500	2e-4
$ \tau=0.001,d=1,N=20 $	7.9688	4e-5	11.4063	2e-4
$ \tau=0.001,d=2,N=10 $	8.0313	6e-4	10.6250	2e-4
$ \tau=0.0005,d=2,N=15 $	73.1250	3e-4	99.9844	1e-4

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Table 2. Comparison of CPU time and error with different $ d $ and $ N $ when $ \theta = 1 $

	$ M_1 $		$ M_2 $
	CPU	$ \\|{\rm{error}}\; \\| $	CPU	$ \\|{\rm{error}}\\|\; $
$ \tau=0.005,d=1,N=15 $	1.0938	5e-5	3.0156	6e-4
$ \tau=0.005,d=1,N=18 $	1.6094	1.5e-4	6.7031	4e-4
$ \tau=0.005,d=1,N=20 $	1.9531	4e-4	8.9650	4e-4
$ \tau=0.005,d=2,N=10 $	2.5608	4e-4	9.3750	6e-4
$ \tau=0.005,d=2,N=12 $	4.9688	3e-4	23.8964	23.8964

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Table 3. Comparison of CPU time and error with different $ d $ and $ N $ when $ \theta = 1 $

	$ M_1 $		$ M_2 $
	CPU	$ \\|{\rm{error}}\; \\| $	CPU	$ \\|{\rm{error}}\\|\; $
$ \tau=0.005,d=1,N=15 $	1.1875	8e-4	3.1875	6e-4
$ \tau=0.005,d=1,N=18 $	1.719	8e-4	6.3594	6e-4
$ \tau=0.005,d=1,N=20 $	2.1875	6e-4	9.3750	6e-4
$ \tau=0.005,d=2,N=10 $	3.4375	1.2e-4	9.5196	6e-4
$ \tau=0.005,d=2,N=12 $	5.2675	4e-4	21.6094	4e-4

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