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

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  • 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.

    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

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

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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