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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.

 Citation: • • 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

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

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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Tables(3)

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