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System of generalized mixed nonlinear ordered variational inclusions
$ \theta $ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations
School of Mathematics, Jilin Normal University, Siping 136000, P. R. China |
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.
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. |
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. |
[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. |


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CPU | CPU | |||
1.1719 | 1.2e-4 | 1.6094 | 4e-4 | |
6.4844 | 1e-5 | 8.2500 | 2e-4 | |
7.9688 | 4e-5 | 11.4063 | 2e-4 | |
8.0313 | 6e-4 | 10.6250 | 2e-4 | |
73.1250 | 3e-4 | 99.9844 | 1e-4 |
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||||
CPU | CPU | |||
1.1719 | 1.2e-4 | 1.6094 | 4e-4 | |
6.4844 | 1e-5 | 8.2500 | 2e-4 | |
7.9688 | 4e-5 | 11.4063 | 2e-4 | |
8.0313 | 6e-4 | 10.6250 | 2e-4 | |
73.1250 | 3e-4 | 99.9844 | 1e-4 |
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||||
CPU | CPU | |||
1.0938 | 5e-5 | 3.0156 | 6e-4 | |
1.6094 | 1.5e-4 | 6.7031 | 4e-4 | |
1.9531 | 4e-4 | 8.9650 | 4e-4 | |
2.5608 | 4e-4 | 9.3750 | 6e-4 | |
4.9688 | 3e-4 | 23.8964 | 23.8964 |
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CPU | CPU | |||
1.0938 | 5e-5 | 3.0156 | 6e-4 | |
1.6094 | 1.5e-4 | 6.7031 | 4e-4 | |
1.9531 | 4e-4 | 8.9650 | 4e-4 | |
2.5608 | 4e-4 | 9.3750 | 6e-4 | |
4.9688 | 3e-4 | 23.8964 | 23.8964 |
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||||
CPU | CPU | |||
1.1875 | 8e-4 | 3.1875 | 6e-4 | |
1.719 | 8e-4 | 6.3594 | 6e-4 | |
2.1875 | 6e-4 | 9.3750 | 6e-4 | |
3.4375 | 1.2e-4 | 9.5196 | 6e-4 | |
5.2675 | 4e-4 | 21.6094 | 4e-4 |
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||||
CPU | CPU | |||
1.1875 | 8e-4 | 3.1875 | 6e-4 | |
1.719 | 8e-4 | 6.3594 | 6e-4 | |
2.1875 | 6e-4 | 9.3750 | 6e-4 | |
3.4375 | 1.2e-4 | 9.5196 | 6e-4 | |
5.2675 | 4e-4 | 21.6094 | 4e-4 |
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