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Article Contents

# Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations

• * Corresponding author: Xian-Ming Gu

The first author is supported by the Fundamental Research Funds for the Central Universities (JBK2101012). The second author is supported by NSFC (11801463) and the Applied Basic Research Project of Sichuan Province (2020YJ0007)

• In this work, two fully novel finite difference schemes for two-dimensional time-fractional mixed diffusion and diffusion-wave equation (TFMDDWEs) are presented. Firstly, a Hermite and Newton quadratic interpolation polynomial have been used for time discretization and central quotient has used in spatial direction. The H2N2 finite difference is constructed. Secondly, in order to increase computational efficiency, the sum-of-exponential is used to approximate the kernel function in the fractional-order operator. The fast H2N2 finite difference is obtained. Thirdly, the stability and convergence of two schemes are studied by energy method. When the tolerance error $\epsilon$ of fast algorithm is sufficiently small, it proves that both of difference schemes are of $3-\beta\; (1<\beta<2)$ order convergence in time and of second order convergence in space. Finally, numerical results demonstrate the theoretical convergence and effectiveness of the fast algorithm.

Mathematics Subject Classification: Primary: 65M06; 65M12; Secondary: 65M15.

 Citation:

• Table 1.  Comparison of the temporal convergence order and elapsed CPU time of both DS (25)-(27) and FS (54)-(60) for Example 5.1 with different $(\alpha,\beta),$ $M=\lceil N^{\frac{3-\beta}{2}}\rceil$ and $\epsilon=10^{-12}$

 $\beta$ $N$ $\alpha=0.3$ $E(M,N)$(DS) $E(M,N)$(FA) $R_\tau$(DS) $R_\tau$(FA) CPU(DA) CPU(FA) 1280 1.18e-5 1.10e-5 – – 5.84 3.70 1.3 2560 3.62e-6 3.43e-6 1.70 1.68 18.22 6.13 5120 1.11e-6 1.06e-6 1.70 1.69 76.42 22.10 10240 3.42e-7 3.32e-7 1.70 1.67 319.5 76.91 $1.5$ 1280 5.11e-5 4.95e-5 – – 3.75 2.88 2560 1.80e-5 1.75e-5 1.51 1.50 12.83 7.141 5120 6.34e-6 6.19e-6 1.51 1.50 44.41 14.91 10240 2.24e-6 2.18e-6 1.50 1.50 160.5 32.34 $1.7$ 1280 2.36e-4 2.26e-4 – – 4.93 1.94 2560 9.52e-5 9.10e-5 1.31 1.31 12.44 3.47 5120 3.87e-5 3.71e-5 1.30 1.30 34.97 11.20 10240 1.57e-5 1.50e-5 1.30 1.30 112.4 25.5 $\beta$ $N$ $\alpha=0.9$ $E(M,N)$(DS) $E(M,N)$(FA) $R_\tau$(DS) $R_\tau$(FA) CPU(DA) CPU(FA) 1.3 1280 1.13e-5 1.06e-5 – – 5.84 4.094 2560 3.48e-6 3.29e-6 1.70 1.68 19.45 8.906 5120 1.07e-6 1.02e-6 1.70 1.69 77.95 26.44 10240 3.29e-7 3.17e-7 1.70 1.69 301.6 59.34 $1.5$ 1280 4.91e-5 4.79e-5 – – 4.22 2.09 2560 1.73e-5 1.69e-5 1.51 1.50 13.23 10.00 5120 6.09e-6 5.99e-6 1.51 1.50 42.39 13.30 10240 2.15e-6 2.12e-6 1.50 1.50 159.8 30.58 $1.7$ 1280 2.21e-4 2.16e-4 – – 3.56 2.06 2560 8.88e-5 8.70e-5 1.31 1.31 9.20 3.45 5120 3.62e-5 3.54e-5 1.30 1.30 30.73 12.98 10240 1.46e-5 1.44e-5 1.30 1.30 116.9 25.63

Table 2.  Comparison of the spatial convergence order and elapsed CPU time for implementing the DS (25)-(27) and FS (54)-(60) with different $(\alpha,\beta),$ $N = 30000$, $\epsilon = 10^{-12}$ (Example 5.1)

 $\beta$ $M$ $\alpha=0.3$ $\alpha=0.9$ $E(M,N)$ $R_M$ CPU(DA) CPU(FA) $E(M,N)$ $R_M$ CPU(DA) CPU(FA) 16 8.35e-3 - 512 7.09 8.01e-3 - 680.7 3.84 1.3 32 2.09e-3 2.00 528 4.89 2.00e-3 2.00 717.3 4.64 64 5.22e-4 2.00 553 4.97 5.00e-4 2.00 788.5 5.42 128 1.30e-4 2.00 668 6.78 1.25e-4 2.00 898.0 7.14 1.5 16 8.13e-3 - 407.6 2.47 7.82e-3 - 420.2 4.08 32 2.03e-3 2.00 419.9 3.05 1.96e-3 2.00 412.4 4.64 64 5.07e-4 2.00 448.1 3.58 4.89e-4 2.00 424.8 5.69 128 1.27e-4 2.00 471.7 5.00 1.22e-4 2.00 456.4 7.31 1.7 16 8.070e-3 - 567.4 4.00 7.72-3 - 565.1 3.91 32 2.016e-3 2.00 556.8 5.02 1.93-3 2.00 556.2 4.61 64 5.045e-4 2.00 543.1 4.84 4.82-4 2.00 555.2 4.89 128 1.267e-4 1.99 709.9 7.11 1.21-4 1.99 822.5 7.34

Table 3.  Comparison of the temporal convergence order and elapsed CPU time of DS (25)-(27) and FS (54)-(60) for Example 5.2 with different $(\alpha,\beta),$ $M = \lceil N^{\frac{3-\beta}{2}}\rceil$, $\epsilon = 10^{-12}$

 $\beta$ $N$ $\alpha=0.3$ $\alpha=0.9$ $M$ $E(M,N)$ $R_M$ CPU(DA) CPU(FA) $E(M,N)$ $R_M$ CPU(DA) CPU(FA) 256 111 1.320e-2 - 23.42 24.41 1.284e-2 - 22.72 59.27 1.3 512 200 4.141e-3 1.67 221.3 112.7 4.030e-3 1.67 205.9 349.6 1024 362 1.282e-3 1.69 1658 532.1 1.248e-3 1.69 1627 1305 2048 652 3.998e-4 1.68 17970 4399 3.894e-4 1.68 18860 8096 1.5 256 64 4.217e-2 - 2.797 9.219 4.118e-2 - 3.781 7.797 512 107 1.523e-2 1.47 70.28 120.3 1.487e-2 1.47 72.05 134.5 1024 181 5.359e-3 1.51 603.3 583.5 5.234e-3 1.51 596.8 644.8 2048 304 1.908e-3 1.49 5368 2444 1.864e-3 1.49 5280 2514 1.7 256 36 1.341e-1 - 0.6563 4.859 1.335e-1 - 0.7656 3.656 512 57 5.377e-2 1.32 4.859 8.156 5.352e-2 1.32 4.469 7.516 1024 90 2.164e-2 1.31 25.58 22.64 2.154e-2 1.31 19.25 27.47 2048 142 8.712e-3 1.31 1604 833.1 8.674e-3 1.31 411.0 859.8

Table 4.  Comparison of the spatial convergence order and elapsed CPU time of the DS (25)-(27) and FS (54)-(60) for Example 5.2 with different $(\alpha,\beta),\;$ $N = 20000,\;$ $\epsilon = 10^{-12}$

 $\beta$ $M$ $\alpha=0.3$ $\alpha=0.9$ $E(M,N)$ $R_M$ CPU(DA) CPU(FA) $E(M,N)$ $R_M$ CPU(DA) CPU(FA) 8 2.744e+0 - 546.8 20.41 2.673e+0 - 495.1 22.78 1.3 16 6.831e-1 2.01 591.7 19.48 6.656e-1 2.01 596.5 22.00 32 1.706e-1 2.00 1055 33.09 1.662e-1 2.00 1061 35.67 64 4.264e-2 2.00 2978 207.0 4.155e-2 2.00 3099 211.3 1.5 8 2.686e+0 - 444.3 19.80 2.619e+0 - 447.4 19.89 16 6.688e-1 2.01 522.7 19.92 6.522e-1 2.01 526.1 20.88 32 1.670e-1 2.00 716.8 33.13 1.629e-1 2.00 726.5 34.25 64 4.175e-2 2.00 2039 200.8 4.071e-2 2.00 2084 224.2 1.7 8 2.554e+0 - 496.5 21.30 2.537e+0 - 598.6 20.52 16 6.358e-1 2.01 653.1 22.45 6.316e-1 2.01 655.5 21.67 32 1.588e-1 2.00 1124 36.06 1.578e-1 2.00 875.2 36.80 64 3.972e-2 2.00 3042 229.2 3.946e-2 2.00 2092 233.5
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