February  2022, 27(2): 1179-1207. doi: 10.3934/dcdsb.2021086

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

School of Economic Mathematics/Institute of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China

* Corresponding author: Xian-Ming Gu

Received  May 2020 Revised  October 2020 Published  February 2022 Early access  March 2021

Fund Project: 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.

Citation: Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1179-1207. doi: 10.3934/dcdsb.2021086
References:
[1]

L. Banjai and M. López-Fernández, Efficient high order algorithms for fractional integrals and fractional differential equations, Numer. Math., 141 (2019), 289-317.  doi: 10.1007/s00211-018-1004-0.

[2]

D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, Ph.D. dissertation, University of Nevada, Reno, 1998.

[3]

B. L. BuzbeeG. H. Golub and C. W. Nielson, On direct methods for solving Poisson's equations, SIAM J. Numer. Anal., 7 (1970), 627-656.  doi: 10.1137/0707049.

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M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent-Ⅱ, Geothys. J. Int., 13 (1967), 529-539. 

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M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophy., 91 (1971), 134-147.  doi: 10.1007/BF00879562.

[6]

A. Cartea and D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Phys. A., 374 (2007), 749-763. 

[7]

H. ChenS. Lü and W. Chen, A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients, J. Comput. Appl. Math., 330 (2018), 380-397.  doi: 10.1016/j.cam.2017.09.011.

[8]

J. ChenF. LiuV. AnhS. ShenQ. Liu and C. Liao, The analytical solution and numerical solution of the fractional diffusion-wave equation with damping, Appl. Math. Comput., 219 (2012), 1737-1748.  doi: 10.1016/j.amc.2012.08.014.

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W. Chen, A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos, 16 (2006), 023126, 6 pages. doi: 10.1063/1.2208452.

[10]

V. Daftardar-Gejji and S. Bhalekar, Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method, Appl. Math. Comput., 202 (2008), 113-120.  doi: 10.1016/j.amc.2008.01.027.

[11]

L. FengF. Liu and I. Turner, Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains, Commun. Nonlinear Sci. Numer. Simulat., 70 (2019), 354-371.  doi: 10.1016/j.cnsns.2018.10.016.

[12]

L. FengF. LiuI. Turner and L. Zheng, Nmerical methods and analysis for simulating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigid plates, Int. J. Heat Mass Transf., 21 (2018), 1073-1103. 

[13]

C. FetecauM. Athar and C. Fetecau, Unsteady flow of a generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate, Comput. Math. Appl., 57 (2009), 596-603.  doi: 10.1016/j.camwa.2008.09.052.

[14]

R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191. 

[15]

X.-M. Gu and S.-L. Wu, A parallel-in-time iterative algorithm for Volterra partial integral-differential problems with weakly singular kernel, J. Comput. Phys., 417 (2020), 109576, 17 pages. doi: 10.1016/j.jcp.2020.109576.

[16]

Z.-P. Hao and G. Lin, Finite difference schemes for multi-term time-fractional mixed diffusion-wave equations, preprint, arXiv: 1607.07104.

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H. JiangF. LiuI. Turner and K. Burrage, Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl., 64 (2012), 3377-3388.  doi: 10.1016/j.camwa.2012.02.042.

[18]

S. Jiang, J. Zhang, Q. Zhang and Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. doi: 10.4208/cicp.OA-2016-0136.

[19]

R. KeM. K. Ng and H.-W. Sun, A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations, J. Comput. Phys., 303 (2015), 203-211.  doi: 10.1016/j.jcp.2015.09.042.

[20]

J. F. KellyR. J. McGough and M. M. Meerschaert, Analytical time-domain Green's functions for power-law media, J. Acoust. Soc. Am., 124 (2008), 2861-2872.  doi: 10.1121/1.2977669.

[21]

A. La PortaG. A. VothA. M. CrawfordJ. Alexander and E. Bodenschatz, Fluid particle accelerations in fully developed turbulence, Nature, 409 (2001), 1017-1019.  doi: 10.1038/35059027.

[22]

J. R. Li, A fast time stepping method for evaluating fractional integrals, SIAM J. Sci. Comput., 31 (2009/10), 4696-4714.  doi: 10.1137/080736533.

[23]

Z. LiuF. Liu and F. Zeng, An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations, Appl. Numer. Math., 136 (2019), 139-151.  doi: 10.1016/j.apnum.2018.10.005.

[24]

M. López-FernándezC. Lubich and A. Schädle, Adaptive, fast, and oblivious convolution in evolution equations with memory, SIAM J. Sci. Comput., 30 (2008), 1015-1037.  doi: 10.1137/060674168.

[25]

J. C. López Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal., 27 (1990), 20-31.  doi: 10.1137/0727002.

[26]

X. LuH. K. Pang and H. W. Sun, Fast approximate inversion of a block triangular Toeplitz matrix with applications to fractional sub-diffusion equations, Numer. Linear Algebra Appl., 22 (2015), 866-882.  doi: 10.1002/nla.1972.

[27]

Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548.  doi: 10.1016/j.jmaa.2010.08.048.

[28]

W. McLean, Exponential sum approximations for $t^{-\beta}$, in Contemporary Computational Mathematics–a Celebration of the 80th Birthday of Ian Sloan (eds. J. Dick, F. Kuo, and H. Woźniakowski), Springer, Cham, Switzerland, (2018), 911–930.

[29]

E. ScalasR. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance, Phys. A., 284 (2000), 376-384.  doi: 10.1016/S0378-4371(00)00255-7.

[30]

E. Scalas, R. Gorenflo and F. Mainardi, Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation, Phys. Rev. E., 69 (2004), 011107, 8 pages. doi: 10.1103/PhysRevE.69.011107.

[31]

W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), 134-144.  doi: 10.1063/1.528578.

[32]

J. Shen, C. Li and Z.-Z. Sun, An H2N2 interpolation for Caputo derivative with order in (1, 2) and its application to time fractional hyperbolic equation in more than one space dimension, J. Sci. Comput., 83 (2020), Paper No. 38, 29 pp. doi: 10.1007/s10915-020-01219-8.

[33]

P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298.  doi: 10.1115/1.3167615.

[34]

D. VieruC. Fetecau and C. Fetecau, Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate, Appl. Math. Comput., 200 (2008), 459-464.  doi: 10.1016/j.amc.2007.11.017.

[35]

Z. WangX. Huang and G. Shi, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Comput. Math. Appl., 62 (2011), 1531-1539.  doi: 10.1016/j.camwa.2011.04.057.

[36]

Y. WeiY. ZhaoY. TangW. LingZ. Shi and K. Li, High-accuracy analysis of finite element method for two-term mixed time-fractional diffusion-wave equations, Sci. China. Inform., 48 (2018), 871-887.  doi: 10.1360/N112017-00295.

[37]

Z. YongD. A. BensonM. M. Meerschaert and H.-P. Scheffler, On using random walks to solve the space-fractional advection-dispersion equations, J. Stat. Phys., 123 (2006), 89-110.  doi: 10.1007/s10955-006-9042-x.

[38]

Y. YuP. Perdikaris and G. E. Karniadakis, Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms, J. Comput. Phys., 323 (2016), 219-242.  doi: 10.1016/j.jcp.2016.06.038.

[39]

F. Zeng, I. Turner, K. Burrage and G. E. Karniadakis, A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations, SIAM J. Sci. Comput., 40 (2018), A2986–A3011. doi: 10.1137/18M1168169.

[40]

F. ZengZ. Zhang and G. E. Karniadakis, Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations, J. Comput. Phys., 307 (2016), 15-33.  doi: 10.1016/j.jcp.2015.11.058.

[41]

Y. ZhaoF. WangX. HuZ. Shi and Y. Tang, Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain, Comput. Math. Appl., 78 (2019), 1705-1719.  doi: 10.1016/j.camwa.2018.11.028.

show all references

References:
[1]

L. Banjai and M. López-Fernández, Efficient high order algorithms for fractional integrals and fractional differential equations, Numer. Math., 141 (2019), 289-317.  doi: 10.1007/s00211-018-1004-0.

[2]

D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, Ph.D. dissertation, University of Nevada, Reno, 1998.

[3]

B. L. BuzbeeG. H. Golub and C. W. Nielson, On direct methods for solving Poisson's equations, SIAM J. Numer. Anal., 7 (1970), 627-656.  doi: 10.1137/0707049.

[4]

M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent-Ⅱ, Geothys. J. Int., 13 (1967), 529-539. 

[5]

M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophy., 91 (1971), 134-147.  doi: 10.1007/BF00879562.

[6]

A. Cartea and D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Phys. A., 374 (2007), 749-763. 

[7]

H. ChenS. Lü and W. Chen, A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients, J. Comput. Appl. Math., 330 (2018), 380-397.  doi: 10.1016/j.cam.2017.09.011.

[8]

J. ChenF. LiuV. AnhS. ShenQ. Liu and C. Liao, The analytical solution and numerical solution of the fractional diffusion-wave equation with damping, Appl. Math. Comput., 219 (2012), 1737-1748.  doi: 10.1016/j.amc.2012.08.014.

[9]

W. Chen, A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos, 16 (2006), 023126, 6 pages. doi: 10.1063/1.2208452.

[10]

V. Daftardar-Gejji and S. Bhalekar, Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method, Appl. Math. Comput., 202 (2008), 113-120.  doi: 10.1016/j.amc.2008.01.027.

[11]

L. FengF. Liu and I. Turner, Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains, Commun. Nonlinear Sci. Numer. Simulat., 70 (2019), 354-371.  doi: 10.1016/j.cnsns.2018.10.016.

[12]

L. FengF. LiuI. Turner and L. Zheng, Nmerical methods and analysis for simulating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigid plates, Int. J. Heat Mass Transf., 21 (2018), 1073-1103. 

[13]

C. FetecauM. Athar and C. Fetecau, Unsteady flow of a generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate, Comput. Math. Appl., 57 (2009), 596-603.  doi: 10.1016/j.camwa.2008.09.052.

[14]

R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), 167-191. 

[15]

X.-M. Gu and S.-L. Wu, A parallel-in-time iterative algorithm for Volterra partial integral-differential problems with weakly singular kernel, J. Comput. Phys., 417 (2020), 109576, 17 pages. doi: 10.1016/j.jcp.2020.109576.

[16]

Z.-P. Hao and G. Lin, Finite difference schemes for multi-term time-fractional mixed diffusion-wave equations, preprint, arXiv: 1607.07104.

[17]

H. JiangF. LiuI. Turner and K. Burrage, Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl., 64 (2012), 3377-3388.  doi: 10.1016/j.camwa.2012.02.042.

[18]

S. Jiang, J. Zhang, Q. Zhang and Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. doi: 10.4208/cicp.OA-2016-0136.

[19]

R. KeM. K. Ng and H.-W. Sun, A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations, J. Comput. Phys., 303 (2015), 203-211.  doi: 10.1016/j.jcp.2015.09.042.

[20]

J. F. KellyR. J. McGough and M. M. Meerschaert, Analytical time-domain Green's functions for power-law media, J. Acoust. Soc. Am., 124 (2008), 2861-2872.  doi: 10.1121/1.2977669.

[21]

A. La PortaG. A. VothA. M. CrawfordJ. Alexander and E. Bodenschatz, Fluid particle accelerations in fully developed turbulence, Nature, 409 (2001), 1017-1019.  doi: 10.1038/35059027.

[22]

J. R. Li, A fast time stepping method for evaluating fractional integrals, SIAM J. Sci. Comput., 31 (2009/10), 4696-4714.  doi: 10.1137/080736533.

[23]

Z. LiuF. Liu and F. Zeng, An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations, Appl. Numer. Math., 136 (2019), 139-151.  doi: 10.1016/j.apnum.2018.10.005.

[24]

M. López-FernándezC. Lubich and A. Schädle, Adaptive, fast, and oblivious convolution in evolution equations with memory, SIAM J. Sci. Comput., 30 (2008), 1015-1037.  doi: 10.1137/060674168.

[25]

J. C. López Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal., 27 (1990), 20-31.  doi: 10.1137/0727002.

[26]

X. LuH. K. Pang and H. W. Sun, Fast approximate inversion of a block triangular Toeplitz matrix with applications to fractional sub-diffusion equations, Numer. Linear Algebra Appl., 22 (2015), 866-882.  doi: 10.1002/nla.1972.

[27]

Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548.  doi: 10.1016/j.jmaa.2010.08.048.

[28]

W. McLean, Exponential sum approximations for $t^{-\beta}$, in Contemporary Computational Mathematics–a Celebration of the 80th Birthday of Ian Sloan (eds. J. Dick, F. Kuo, and H. Woźniakowski), Springer, Cham, Switzerland, (2018), 911–930.

[29]

E. ScalasR. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance, Phys. A., 284 (2000), 376-384.  doi: 10.1016/S0378-4371(00)00255-7.

[30]

E. Scalas, R. Gorenflo and F. Mainardi, Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation, Phys. Rev. E., 69 (2004), 011107, 8 pages. doi: 10.1103/PhysRevE.69.011107.

[31]

W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), 134-144.  doi: 10.1063/1.528578.

[32]

J. Shen, C. Li and Z.-Z. Sun, An H2N2 interpolation for Caputo derivative with order in (1, 2) and its application to time fractional hyperbolic equation in more than one space dimension, J. Sci. Comput., 83 (2020), Paper No. 38, 29 pp. doi: 10.1007/s10915-020-01219-8.

[33]

P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298.  doi: 10.1115/1.3167615.

[34]

D. VieruC. Fetecau and C. Fetecau, Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate, Appl. Math. Comput., 200 (2008), 459-464.  doi: 10.1016/j.amc.2007.11.017.

[35]

Z. WangX. Huang and G. Shi, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Comput. Math. Appl., 62 (2011), 1531-1539.  doi: 10.1016/j.camwa.2011.04.057.

[36]

Y. WeiY. ZhaoY. TangW. LingZ. Shi and K. Li, High-accuracy analysis of finite element method for two-term mixed time-fractional diffusion-wave equations, Sci. China. Inform., 48 (2018), 871-887.  doi: 10.1360/N112017-00295.

[37]

Z. YongD. A. BensonM. M. Meerschaert and H.-P. Scheffler, On using random walks to solve the space-fractional advection-dispersion equations, J. Stat. Phys., 123 (2006), 89-110.  doi: 10.1007/s10955-006-9042-x.

[38]

Y. YuP. Perdikaris and G. E. Karniadakis, Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms, J. Comput. Phys., 323 (2016), 219-242.  doi: 10.1016/j.jcp.2016.06.038.

[39]

F. Zeng, I. Turner, K. Burrage and G. E. Karniadakis, A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equations, SIAM J. Sci. Comput., 40 (2018), A2986–A3011. doi: 10.1137/18M1168169.

[40]

F. ZengZ. Zhang and G. E. Karniadakis, Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations, J. Comput. Phys., 307 (2016), 15-33.  doi: 10.1016/j.jcp.2015.11.058.

[41]

Y. ZhaoF. WangX. HuZ. Shi and Y. Tang, Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain, Comput. Math. Appl., 78 (2019), 1705-1719.  doi: 10.1016/j.camwa.2018.11.028.

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
$ \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
$ \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
$ \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
$ \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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