$ \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: April 30, 2018

Revised: February 28, 2019

Published: November 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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References

[1]	M. Chen and R. Temam, Incremental unknowns for solving partial differential equations, Numer. Math., 59 (1991), 255-251. doi: 10.1007/BF01385779.
[2]	M. Chen and R. Temam, Incremental unknowns in finite differences: Condition number of the matrix, SIAM J. Matrix Anal. Appl., 14 (1993), 432-455. doi: 10.1137/0614031.
[3]	M. Chen and R. Temam, Nonlinear Galerkin method with multilevel incremental unknowns, Contributions in Numerical Mathematics (ed. R. P. Agarwal), WSSIAA2 (1993), 151–164. doi: 10.1142/9789812798886_0012.
[4]	A. Eden, B. Michaux and J. M. Rakotoson, Doubly nonlinear parabolic-type equations as dynamical systems, J. Dyna. Diff. Equ., 3 (1991), 87-129. doi: 10.1007/BF01049490.
[5]	C. Foias, O. Manley and R. Temam, Modeling of the interaction of small and large eddies in two-dimensional turbulent flow, Math. Modelling Numer. Anal., 22 (1988), 93-114. doi: 10.1051/m2an/1988220100931.
[6]	G. H. Golub and C. F. Van Loan, Matrix Computations, Post and Telecome Press, 2009.
[7]	P. Poullet and A. Boag, Equation-based interpolation and incremental unknowns for solving the three-dimensional Helmholtz equation, Appl. Math. Comput., 232 (2014), 1200-1208. doi: 10.1016/j.amc.2014.01.084.
[8]	L. J. Song and Y. J. Wu, Nonlinear stability of reaction-diffusion equations using wavelet-like incremental unknowns, Appl. Numer. Math., 68 (2013), 83-107. doi: 10.1016/j.apnum.2012.12.003.
[9]	R. Temam, Inertial manifolds and multigrid methods, SIAM J. Math. Anal., 21 (1990), 154-178. doi: 10.1137/0521009.
[10]	Y. Wang, Y. J. Wu and X. Y. Fan, Two parameter preconditioned NSS methods for non-Hermitian and positive definite linear systems, Communication on Applied Mathematics and Computation, 27 (2014), 322-340.
[11]	Y. J. Wu and A. L. Yang, Incremental unknowns for the heat equation with time-dependent coefficients: semi-implicit $\theta$ schemes and their staility, J. Comput. Math., 25 (2007), 573-582.
[12]	Y. J. Wu, Y. Wang, M. L. Zeng and A. L. Yang, Implementation of a modified Marder-Weitzner method for solving nonlinear eigenvalue problems, J. Comput. Appl. Math., 226 (2009), 166-176. doi: 10.1016/j.cam.2008.05.034.
[13]	Y. J. Wu, X. X. Jia and A. L. She, Semi-implicit schemes with multilevel wavelet-like incremental unknowns for solving reaction diffusion equation, Hokkaido Mathematical Journal, 36 (2007), 711-728. doi: 10.14492/hokmj/1272848029.
[14]	A. L. Yang and Y. J. Wu, Wavelet-like block incremental unknowns for numerical computation of anisotropic parabolic equations, World Congress on Computer Science and Information Engineering, 2 (2009), 550-554.
[15]	A. L. Yang and Y. J. Wu, Preconditioning analysis of the one dimensional incremental unknowns method on nonuniform meshes, J. Appl. Math. Comput., 44 (2014), 379-395. doi: 10.1007/s12190-013-0698-5.
[16]	A. L. Yang, Y. J. Wu, Z. D. Huang and J. Y. Yuan, Preconditioning analysis of nonuniform incremental unknowns method for two dimensional elliptic problems, Appl. Math. Modelling, 39 (2015), 5436-5451. doi: 10.1016/j.apm.2015.01.009.
[17]	A. L. Yang, L. J. Song and Y. J. Wu, Algebraic preconditioning analysis of the multilevel block incremental unknowns method for anisotropic elliptic operators, Math. Comput. Modelling, 57 (2013), 512-524. doi: 10.1016/j.mcm.2012.06.031.