Local discontinuous Galerkin schemes for an ultrasonic propagation equation with fractional attenuation

Can Li; Min-Min Li; Zine El Abiddine Fellah

doi:10.3934/dcdsb.2023063

Article Contents

2023, Volume 28, Issue 11: 5494-5513. Doi: 10.3934/dcdsb.2023063

This issue Previous Article Periodic oscillations in the restricted hip-hop $ 2N+1 $-body problem Next Article Well-posedness and finite element approximation for the steady-state closed-loop geothermal system

Local discontinuous Galerkin schemes for an ultrasonic propagation equation with fractional attenuation

1.
Department of Applied Mathematics, Xi'an University of Technology, Xi'an, Shaanxi 710054, China
2.
Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Laboratory of Mechanics and Acoustics, CEDEX 20, 13402 Marseille, France

^*Corresponding author: Can Li
^*Corresponding author: Can Li

Received: October 2022

Revised: February 2023

Early access: March 2023

Published: November 2023

This work is supported by the Science Basic Research Plan in Shaanxi Province of China under Grant No.2023-JC-YB-045.

Abstract / Introduction Full Text(HTML) Figure(0) / Table(4) Related Papers Cited by

Abstract

The goal of this article is to develop local discontinuous Galerkin (LDG) schemes for solving a time fractional equation describing the ultrasonic wave in a homogeneous isotropic porous material. Two novel semi-discrete LDG schemes are designed for the considered model. The semi-discrete LDG schemes are constructed by splitting the original model into a coupled system. The first semi-discrete scheme follows the traditional LDG method by splitting second-order space derivative. The second one splits the original model for both time and space derivatives. The discontinuous Galerkin is used for the spatial discretization. Two kinds of fully discrete LDG schemes are presented by using the Grünwald-Letnikov and L1 approximation formulas for the time fractional derivatives. The $ L^2 $ norm stability and convergence analysis are carried out for both semi-discrete and fully discrete LDG schemes. The stability analysis reveals that the numerical schemes are unconditionally stable in $ L^2 $ norm and convergence with optimal convergence rate. Finally, numerical examples are presented to test the effectiveness of the proposed schemes and the correctness of the theoretical analysis.

Keywords:

Mathematics Subject Classification: Primary: 34A08, 74S25; Secondary: 26A33.

Citation:

Full Text(HTML)

Table 1. The $ L^2 $ errors and convergence orders of scheme (46) and scheme (51) for $ P^{1} $ element with $ h = 1/2000, \gamma = 3 $

$ \tau $	Scheme (46)		Scheme (51)
$ \tau $	$ \\|\cdot\\| $-error	order	$ \\|\cdot\\| $-error	order
1/5	5.0164e-02	$ \ast $	3.4866e-04	$ \ast $
1/10	2.4845e-02	1.0137	1.2763e-04	1.4499
1/20	1.1776e-02	1.0771	3.2934e-05	1.9542
1/40	5.6086e-03	1.0701	8.3057e-06	1.9874
1/80	2.7134e-03	1.0475	2.1335e-06	1.9609

| Show Table

DownLoad: CSV

Table 2. The $ L^2 $ errors and convergence orders of scheme (46) for different $ P^{k} $ elements with $ \tau = h^{k+1}, \gamma = 3 $

$ h $	$ P^{0} $		$ P^{1} $		$ P^{2} $
$ h $	$ \\|\cdot\\| $-error	order	$ \\|\cdot\\| $-error	order	$ \\|\cdot\\| $-error	order
1/5	2.7810e-01	$ \ast $	6.7699e-02	$ \ast $	6.8549e-03	$ \ast $
1/10	1.3414e-01	1.0518	1.7074e-02	1.9873	8.7451e-04	2.9706
1/20	6.5965e-02	1.0240	4.2764e-03	1.9973	1.0989e-04	2.9925
1/40	3.2726e-02	1.0113	1.0696e-03	1.9994	1.3751e-05	2.9984

| Show Table

DownLoad: CSV

Table 3. The $ L^2 $ errors and convergence orders of scheme (51) for different $ P^{k} $ elements with $ \tau = h^{(k+1)/2}, \gamma = 3 $

$ h $	$ P^{0} $		$ P^{1} $		$ P^{2} $
$ h $	$ \\|\cdot\\| $-error	order	$ \\|\cdot\\| $-error	order	$ \\|\cdot\\| $-error	order
1/5	1.1735e+00	$ \ast $	6.4446e-02	$ \ast $	8.2638e-03	$ \ast $
1/10	5.1205e-01	1.1965	1.6766e-02	1.9426	8.8156e-04	3.2287
1/20	1.7857e-01	1.5198	4.2333e-03	1.9857	1.0884e-04	3.0178
1/40	8.6859e-02	1.0397	1.0609e-03	1.9964	1.3375e-05	3.0249

| Show Table

DownLoad: CSV

Table 4. The $ L^2 $ errors and convergence orders of schemes (46) and (51) for $ P^{1} $ element with $ h = 1/2000, \gamma = 2 $

$ \tau $	Scheme (46)		Scheme (51)
$ \tau $	$ \\|\cdot\\| $-error	order	$ \\|\cdot\\| $-error	order
1/5	1.0574e-02	$ \ast $	3.5121e-04	$ \ast $
1/10	5.8509e-03	0.8537	3.4066e-04	0.0440
1/20	2.7058e-03	1.1126	1.3991e-04	1.2838
1/40	1.2358e-03	1.1306	5.1111e-05	1.4528
1/80	5.7308e-04	1.1087	1.8255e-05	1.4853

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	M. Ahmadinia and Z. Safari, Convergence nanalysis a local discontinuous Galerkin method for tempered fractional convection-diffusion equations, ESAIM Math. Model. Numer. Anal., 54 (2020), 59-78. doi: 10.1051/m2an/2019052.
[2]	A. A. Alikhanov, A priori estimates for solutions of boundary value problem for fractional-order equations, Diff.Eq., 46 (2010), 660-666. doi: 10.1134/S0012266110050058.
[3]	J. F. Allard, Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials, Chapman and Hall, London, 1993.
[4]	W. X. Cao, D. F. Li and Z. M. Zhang, Optimal Superconvergence of energy conserving local discontinuous Galerkin methods for wave equations, Commun. Comput. Phys., 21 (2017), 211-236. doi: 10.4208/cicp.120715.100516a.
[5]	W. Cai, W. Chen, J. Fang and S. Holm, A survey on fractional derivative modeling of power-law frequency-dependent viscous dissipative and scattering attenuation in acoustic wave propagation, Appl. Mech. Rev., 70 (2018), 030802. doi: 10.1115/1.4040402.
[6]	J. M. Carcione, F. Cavallini, F. Mainardi and A. Hanyga, Time-domain modeling of constant-Q seismic waves using fractional derivatives, Pure Appl. Geophys., 159 (2002), 1719-1736.
[7]	B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463. doi: 10.1137/S0036142997316712.
[8]	W. H. Deng and J. S. Hesthaven, Local discontinuous Galerkin methods for fractional diffusion equations, ESAIM Math. Model. Numer. Anal., 47 (2013), 1845-1864. doi: 10.1051/m2an/2013091.
[9]	W. H. Deng and J. S. Hesthaven, Local discontinuous Galerkin methods for fractional ordinary differential equations, BIT Numer. Math., 55 (2015), 976-985. doi: 10.1007/s10543-014-0531-z.
[10]	Y. W. Du, Y. Liu, H. Li, Z. C. Fang and S. He, Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation, J. Comput. Phys., 344 (2017), 108-126. doi: 10.1016/j.jcp.2017.04.078.
[11]	L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.
[12]	Z. E. A. Fellah and C. Depollier, Transient acoustic wave propagation in rigid porous media: A time-domain approach, J. Comput. Acoust., 9 (2001), 1163-1173. doi: 10.1142/S0218396X01000723.
[13]	Z. E. A. Fellah, M. Fellah, W. Lauriks, C. Depollier, J.-Y. Chapelon and Y. C. Angel, Solution in time domain of ultrasonic propagation equation in a porous material, Wave Motion, 38 (2003), 151-163. doi: 10.1016/S0165-2125(03)00045-3.
[14]	Z. E. A. Fellah, F. G. Mitri, M. Fellah, E. Ogam and C. Depollier, Ultrasonic characterization of porous absorbing materials: Inverse problem, J. Sound Vibr., 302 (2007), 746-759.
[15]	R. Garra, Fractional-calculus model for temperature and pressure waves in fluid-saturated porous rocks, Phys. Rev. E, 84 (2011), 036605.
[16]	J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods. Algorithms, Analysis, and Applications, Texts in Applied Mathematics, 54. Springer, New York, 2008. doi: 10.1007/978-0-387-72067-8.
[17]	S. Holm, Waves with Power-Law Attenuation, Springer, Berlin, 2019. doi: 10.1007/978-3-030-14927-7.
[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]	B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221. doi: 10.1093/imanum/dru063.
[20]	X. Ji and H. Z.Tang, High-order accurate Runge-Kutta (local) discontinuous Galerkin methods for one-and two-dimensional fractional diffusion equations, Numer. Math. Theor. Meth. Appl., 5 (2012), 333-358. doi: 10.4208/nmtma.2012.m1107.
[21]	D. L. Johnson, J. Koplik and R. Dashen, Theory of dynamic permeability and tortuosity in fluid-saturated porous media, J. Fluid. Mech., 176 (1987), 379-402.
[22]	J. F. Kelly, R. J. McGough and M. M. Meerschaert, Time-domain 3D Green's functions for power law media, J. Acoust. Soc. Am., 124 (2008), 2861-2872.
[23]	R. M. Kirby and G. E. Karniadakis, Selecting the numerical flux in discontinuous Galerkin methods for diffusion problems, J. Sci. Comput., 22/23 (2005), 385-411. doi: 10.1007/s10915-004-4145-5.
[24]	C. Li, T. G. Zhao, W. H. Deng and Y. J. Wu, Orthogonal spline collocation methods for the subdiffusion equation, J. Comput. Appl. Math., 255 (2014), 517-528. doi: 10.1016/j.cam.2013.05.022.
[25]	C. P. Li and Z. Wang, The local discontinuous Galerkin finite element methods for Caputo-type partial differential equations: Numerical analysis, Appl. Numer.Math., 140 (2019), 1-22. doi: 10.1016/j.apnum.2019.01.007.
[26]	C. P. Li and F. H. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Boca Raton, FL, 2015.
[27]	H.-L. Liao, D. Li and J. Zhang, Sharp error estimate of nonuniform L1 formula for time-fractional reaction-subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1112-1133. doi: 10.1137/17M1131829.
[28]	Y. M. Lin and C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001.
[29]	F. Liu, M. M. Meerschaert, R. J. McGough, P. Zhuang and Q. Liu, Numerical methods for solving the multi-term time-fractional wave diffusion equation, Fract. Calc. Appl. Anal., 16 (2013), 9-25. doi: 10.2478/s13540-013-0002-2.
[30]	F. W. Liu, P. H. Zhuang and Q. X. Liu, The Applications and Numerical Methods of Fractional Differential Equations, Science Press, Beijing, 2015.
[31]	Y. Liu, M. Zhang, H. Li and J. C. Li, High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation, Comput. Math. Appl., 73 (2017), 1298-1314. doi: 10.1016/j.camwa.2016.08.015.
[32]	Ch. Lubich, Discretized fractional calculus, SIAM J Math Anal., 17 (1986), 704-719. doi: 10.1137/0517050.
[33]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models,, Imperial College Press, London, 2010. doi: 10.1142/9781848163300.
[34]	M. M. Meerschaert and R. J. McGough, Attenuated fractional wave equations with anisotropy, ASME J. Vib. Acoust., 136 (2014), 050902.
[35]	J. Q. Murillo and S. B. Yuste, On three explicit difference schemes for fractional diffusion and diffusion-wave equations, Phys. Scr., 136 (2009), 14025-14030.
[36]	K. Mustapha and W. McLean, Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation, Numer. Algorithms, 56 (2010), 159-184. doi: 10.1007/s11075-010-9379-8.
[37]	I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999.
[38]	A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994.
[39]	C.-W. Shu, High order WENO and DG methods for time-dependent convection-dominated PDEs: a brief survey of several recent developments, J. Comput. Phys., 316 (2016), 598-613. doi: 10.1016/j.jcp.2016.04.030.
[40]	X. R. Sun, C. Li and F. Q. Zhao, Local discontinuous Galerkin methods for the time tempered fractional diffusion equation, Appl. Math. Comput., 365 (2020), 124725. doi: 10.1016/j.amc.2019.124725.
[41]	Z. Z. Sun and X. N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193-209. doi: 10.1016/j.apnum.2005.03.003.
[42]	T. L. Szabo, Causal theories and data for acoustic attenuation obeying a frequency power-law, J. Acoust. Soc. Am., 97 (1995), 14-24.
[43]	W. Y. Tian, H. Zhou and W. H. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput., 84 (2015), 1703-1727. doi: 10.1090/S0025-5718-2015-02917-2.
[44]	L. L. Wei, X. D. Zhang, Y. N. He and S. Wang, Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schr ödinger equation, Finite Elem. Anal. Desi., 59 (2012), 28-34. doi: 10.1016/j.finel.2012.03.008.
[45]	D. K. Wilson, V. E. Ostashev, S. L. Collier, N. P. Symons, D. F. Aldridge and D. H. Marlin, Time-domain calculations of sound interactions with outdoor ground surfaces, Applied Acoustics, 68 (2007), 173-200.
[46]	Y. Xing, C.-S. Chou and C.-W. Shu, Energy conserving local discontinuous Galerkin methods for wave propagation problems, Inverse Probl. Imaging, 7 (2013), 967-986. doi: 10.3934/ipi.2013.7.967.
[47]	Q. Xu and J. S. Hesthaven, Discontinuous Galerkin method for fractional convection-diffusion equations, SIAM J. Numer.Anal., 52 (2014), 405-423. doi: 10.1137/130918174.
[48]	Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Comm. Comput. Phys., 7 (2010), 1-46. doi: 10.4208/cicp.2009.09.023.
[49]	M. Zhang, Y. Liu and H. Li, High order local discontinuous Galerkin algorithm with time second-order schemes for the two-dimensional nonlinear fractional diffusion equation, Commun. Appl. Math. Comput., 2 (2020), 613-640. doi: 10.1007/s42967-019-00058-1.
[50]	Q. Zhang, J. W. Zhang, S. D. Jiang and Z. M. Zhang, Numerical solution to a linearized time fractional KdV equation on unbounded domains, Math. comput., 87 (2018), 693-719. doi: 10.1090/mcom/3229.