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December  2019, 9(4): 461-481. doi: 10.3934/naco.2019027

$ \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

Received  May 2018 Revised  March 2019 Published  May 2019

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.

Keywords: Stability, wavelet-like incremental unknowns, porous medium diffusion-type equations, finite difference.
Mathematics Subject Classification: Primary: 65M06, 65P40; Secondary: 65M50.
Citation: Yang Wang, Yi-fu Feng. $ \theta $ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 461-481. doi: 10.3934/naco.2019027
References:
[1]

M. Chen and R. Temam, Incremental unknowns for solving partial differential equations, Numer. Math., 59 (1991), 255-251.  doi: 10.1007/BF01385779.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[6] G. H. Golub and C. F. Van Loan, Matrix Computations, Post and Telecome Press, 2009.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[9]

R. Temam, Inertial manifolds and multigrid methods, SIAM J. Math. Anal., 21 (1990), 154-178.  doi: 10.1137/0521009.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Chen and R. Temam, Incremental unknowns for solving partial differential equations, Numer. Math., 59 (1991), 255-251.  doi: 10.1007/BF01385779.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6] G. H. Golub and C. F. Van Loan, Matrix Computations, Post and Telecome Press, 2009.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[9]

R. Temam, Inertial manifolds and multigrid methods, SIAM J. Math. Anal., 21 (1990), 154-178.  doi: 10.1137/0521009.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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$ 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
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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Download as excel

$ 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
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
Table Options

Download as excel

$ 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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