A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

doi:10.1007/s42967-020-00070-w

Communications on Applied Mathematics and Computation ›› 2021, Vol. 3 ›› Issue (1): 61-90.doi: 10.1007/s42967-020-00070-w

• ORIGINAL PAPER • 上一篇

A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

Mehdi Samiee^1,2, Ehsan Kharazmi^1,2, Mark M. Meerschaert³, Mohsen Zayernouri^1,3

1 Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA;
2 Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA;
3 Department of Probability and Statistics, Michigan State University, East Lansing, MI 48824, USA

收稿日期:2019-07-25 修回日期:2020-03-26 发布日期:2021-03-15
通讯作者: Mohsen Zayernouri, zayern@msu.edu E-mail:zayern@msu.edu
基金资助:
This work was supported by the AFOSR Young Investigator Program (YIP) award (FA9550-17-1-0150), the MURI/ARO (W911NF-15-1-0562), the National Science Foundation Award (DMS-1923201), and the ARO Young Investigator Program Award (W911NF-19-1-0444)

A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

Mehdi Samiee^1,2, Ehsan Kharazmi^1,2, Mark M. Meerschaert³, Mohsen Zayernouri^1,3

1 Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA;
2 Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA;
3 Department of Probability and Statistics, Michigan State University, East Lansing, MI 48824, USA

Received:2019-07-25 Revised:2020-03-26 Published:2021-03-15
Contact: Mohsen Zayernouri, zayern@msu.edu E-mail:zayern@msu.edu
Supported by:
This work was supported by the AFOSR Young Investigator Program (YIP) award (FA9550-17-1-0150), the MURI/ARO (W911NF-15-1-0562), the National Science Foundation Award (DMS-1923201), and the ARO Young Investigator Program Award (W911NF-19-1-0444)

摘要/Abstract

摘要： Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings. In complex multi-fractal anomalous transport phenomena, distributed-order partial differential equations appear as tractable mathematical models, where the underlying derivative orders are distributed over a range of values, hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics. We develop a unified, fast, and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying distributed Sobolev spaces and their equivalent norms, we rigorously prove the well-posedness of the weak formulation, and thereby, we carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

关键词: Distributed Sobolev space, Well-posedness analysis, Discrete inf-sup condition, Spectral convergence, Jacobi poly-fractonomials, Legendre polynomials

Abstract: Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings. In complex multi-fractal anomalous transport phenomena, distributed-order partial differential equations appear as tractable mathematical models, where the underlying derivative orders are distributed over a range of values, hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics. We develop a unified, fast, and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying distributed Sobolev spaces and their equivalent norms, we rigorously prove the well-posedness of the weak formulation, and thereby, we carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

Key words: Distributed Sobolev space, Well-posedness analysis, Discrete inf-sup condition, Spectral convergence, Jacobi poly-fractonomials, Legendre polynomials

中图分类号:

Mehdi Samiee, Ehsan Kharazmi, Mark M. Meerschaert, Mohsen Zayernouri. A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(1): 61-90.

参考文献

1. Abbaszadeh, M., Dehghan, M.: An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer. Algorithms 75(1), 173–211 (2017)
2. Ainsworth, M., Glusa, C.: Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver. Comput. Methods Appl. Mech. Eng. 327, 4–35 (2017)
3. Ammi, M.R.S., Jamiai, I.: Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete Contin. Dyn. Syst. Ser. S 11(1), 103 (2018)
4. Ardakani, A.G.: Investigation of Brewster anomalies in one-dimensional disordered media having Lévy-type distribution. Eur. Phys. J. B 89(3), 76 (2016)
5. Armour, K.C., Marshall, J., Scott, J.R., Donohoe, A., Newsom, E.R.: Southern Ocean warming delayed by circumpolar upwelling and equatorward transport. Nat. Geosci. 9(7), 549–554 (2016)
6. Bazhlekova, E., Bazhlekov, I.: Subordination approach to multi-term time-fractional diffusion-wave equations. J. Comput. Appl. Math. 339, 179–192 (2018)
7. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36(6), 1403–1412 (2000)
8. Chechkin, A., Gorenflo, R., Sokolov, I.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66(4), 046129 (2002)
9. Chen, S., Shen, J., Wang, L.L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)
10. Chen, S., Shen, J., Wang, L.L.: Laguerre functions and their applications to tempered fractional differential equations on infinite intervals. J. Sci. Comput. 74, 1–28 (2017)
11. Cheng, A., Wang, H., Wang, K.: A Eulerian–Lagrangian control volume method for solute transport with anomalous diffusion. Numer. Methods Partial Differ. Equ. 31(1), 253–267 (2015)
12. Coronel-Escamilla, A., Gómez-Aguilar, J., Torres, L., Escobar-Jiménez, R.: A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys. A Stat. Mech. Appl. 491, 406–424 (2018)
13. Duan, B., Jin, B., Lazarov, R., Pasciak, J., Zhou, Z.: Space-time Petrov–Galerkin FEM for fractional diffusion problems. Comput. Methods Appl. Math. 18, 1 (2017)
14. Duan, J.S., Baleanu, D.: Steady periodic response for a vibration system with distributed order derivatives to periodic excitation. J. Vib. Control 21, 3124 (2018)
15. Eab, C., Lim, S.: Fractional Langevin equations of distributed order. Phys. Rev. E 83(3), 031136 (2011)
16. Edery, Y., Dror, I., Scher, H., Berkowitz, B.: Anomalous reactive transport in porous media: experiments and modeling. Phys. Rev. E 91(5), 052130 (2015)
17. Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in ? d. Numer. Methods Partial Differ. Equ. 23(2), 256 (2007)
18. Fan, W., Liu, F.: A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain. Appl. Math. Lett. 77, 114–121 (2018)
19. Gorenflo, R., Luchko, Y., Yamamoto, M.: Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18(3), 799–820 (2015)
20. Goychuk, I.: Anomalous transport of subdiffusing cargos by single kinesin motors: the role of mechano-chemical coupling and anharmonicity of tether. Phys. Biol. 12(1), 016,013 (2015)
21. Iwayama, T., Murakami, S., Watanabe, T.: Anomalous eddy viscosity for two-dimensional turbulence. Phys. Fluids 27(4), 045104 (2015). https://doi.org/10.1063/1.4916956
22. Jin, B., Lazarov, R., Sheen, D., Zhou, Z.: Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data. Fract. Calc. Appl. Anal. 19(1), 69–93 (2016)
23. Jin, B., Lazarov, R., Thomée, V., Zhou, Z.: On nonnegativity preservation in finite element methods for subdiffusion equations. Math. Comput. 86(307), 2239–2260 (2017)
24. Kharazmi, E., Zayernouri, M.: Fractional pseudo-spectral methods for distributed-order fractional PDEs. Int. J. Comput. Math. 95(6/7), 1340–1361 (2018)
25. Kharazmi, E., Zayernouri, M., Karniadakis, G.E.: Petrov–Galerkin and spectral collocation methods for distributed order differential equations. SIAM J. Sci. Comput. 39(3), A1003–A1037 (2017)
26. Kharazmi, E., Zayernouri, M., Karniadakis, G.E.: A Petrov–Galerkin spectral element method for fractional elliptic problems. Comput. Methods Appl. Mech. Eng. 324, 512–536 (2017)
27. Klages, R., Radons, G., Sokolov, I.M.: Anomalous Transport: Foundations and Applications. WileyVCH, New York (2008)
28. Konjik, S., Oparnica, L., Zorica, D.: Distributed-order fractional constitutive stress-strain relation in wave propagation modeling. Zeitsch. Angew. Math. Phys. 70(2), 51 (2019)
29. Li, X., Rui, H.: Two temporal second-order H1-Galerkin mixed finite element schemes for distributedorder fractional sub-diffusion equations. Numer. Algorithms 79(4), 1107–1130 (2018)
30. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)
31. Li, X., Xu, C.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8(5), 1016 (2010)
32. Liao, Hl, Lyu, P., Vong, S., Zhao, Y.: Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations. Numer. Algorithms 75(4), 845–878 (2017)
33. Luchko, Y.: Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12(4), 409–422 (2009)
34. Macías-Díaz, J.: An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions. Commun. Nonlinear Sci. Numer. Simul. 59, 67–87 (2018)
35. Maday, Y.: Analysis of spectral projectors in one-dimensional domains. Math. Comput. 55(192), 537– 562 (1990)
36. Mainardi, F., Mura, A., Gorenflo, R., Stojanović, M.: The two forms of fractional relaxation of distributed order. J. Vib. Control 13(9/10), 1249–1268 (2007)
37. Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R.: Time-fractional diffusion of distributed order. J. Vib. Control 14(9/10), 1267–1290 (2008)
38. Mao, Z., Shen, J.: Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243–261 (2016)
39. Mao, Z., Shen, J.: Spectral element method with geometric mesh for two-sided fractional differential equations. Adv. Comput. Math. 44, 1–27 (2017)
40. Mashelkar, R., Marrucci, G.: Anomalous transport phenomena in rapid external flows of viscoelastic fluids. Rheol. Acta 19(4), 426–431 (1980)
41. Meerschaert, M.M.: Fractional Calculus, Anomalous Diffusion, and Probability. Fractional Dynamics: Recent Advances, pp. 265–284. World Scientific, Singapore (2012)
42. Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus, vol. 43. Walter de Gruyter, Berlin (2012)
43. Metzler, R., Jeon, J.H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014)
44. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)
45. Naghibolhosseini, M.: Estimation of outer-middle ear transmission using DPOAEs and fractionalorder modeling of human middle ear. Ph.D. thesis, City University of New York, NY (2015)
46. Naghibolhosseini, M., Long, G.R.: Fractional-order modelling and simulation of human ear. Int. J. Comput. Math. 95, 1–17 (2017)
47. Perdikaris, P., Karniadakis, G.E.: Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng. 42(5), 1012–1023 (2014)
48. Regner, B.M.: Randomness in biological transport. UC San Diego Electronic Theses and Dissertations, UC San Diego (2014)
49. Samiee, M., Akhavan-Safaei, A., Zayernouri, M.: A fractional subgrid-scale model for turbulent flows: theoretical formulation and a priori study. arXiv:1909.09943 (2019)
50. Samiee, M., Zayernouri, M., Meerschaert, M.M.: A unified spectral method for FPDES with two-sided derivatives; part I: a fast solver. J. Comput. Phys. 385, 225–243 (2019)
51. Samiee, M., Zayernouri, M., Meerschaert, M.M.: A unified spectral method for FPDES with two-sided derivatives; part II: stability, and error analysis. J. Comput. Phys. 385, 244–261 (2019)
52. Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer Science & Business Media, New York (2011)
53. Sokolov, I., Chechkin, A., Klafter, J.: Distributed-order fractional kinetics. arXiv preprint condmat/0401146 (2004)
54. Suzuki, J., Zayernouri, M., Bittencourt, M., Karniadakis, G.: Fractional-order uniaxial visco-elastoplastic models for structural analysis. Comput. Methods Appl. Mech. Eng. 308, 443–467 (2016)
55. Suzuki, J.L., Zayernouri, M.: An automated singularity-capturing scheme for fractional differential equations. arXiv:1810.12219 (2018)
56. Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703–1727 (2015)
57. Tomovski, Ž., Sandev, T.: Distributed-order wave equations with composite time fractional derivative. Int. J. Comput. Math. 95, 1–14 (2017)
58. Tomovski, Ž., Sandev, T.: Distributed-order wave equations with composite time fractional derivative. Int. J. Comput. Math. 95(6/7), 1100–1113 (2018)
59. Tyukhova, A., Dentz, M., Kinzelbach, W., Willmann, M.: Mechanisms of anomalous dispersion in flow through heterogeneous porous media. Phys. Rev. Fluids 1(7), 074,002 (2016)
60. Varghaei, P., Kharazmi, E., Suzuki, J.L., Zayernouri, M.: Vibration analysis of geometrically nonlinear and fractional viscoelastic cantilever beams. arXiv:1909.02142 (2019)
61. Yamamoto, M.: Weak solutions to non-homogeneous boundary value problems for time-fractional diffusion equations. J. Math. Anal. Appl. 460(1), 365–381 (2018)
62. Zaky, M.A.: A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn. 2667, 1–15 (2018)
63. Zayernouri, M., Ainsworth, M., Karniadakis, G.E.: Tempered fractional Sturm–Liouville eigenproblems. SIAM J. Sci. Comput. 37(4), A1777–A1800 (2015)
64. Zayernouri, M., Ainsworth, M., Karniadakis, G.E.: A unified Petrov–Galerkin spectral method for fractional PDES. Comput. Methods Appl. Mech. Eng. 283, 1545–1569 (2015)
65. Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)
66. Zhang, Y., Meerschaert, M.M., Baeumer, B., LaBolle, E.M.: Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme. Water Resour. Res. 51(8), 6311–6337 (2015)
67. Zhang, Y., Meerschaert, M.M., Neupauer, R.M.: Backward fractional advection dispersion model for contaminant source prediction. Water Resour. Res. 52(4), 2462–2473 (2016)

A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

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