An efficient localized collocation solver for anomalous diffusion on surfaces
-
Zhuochao Tang
,
HongGuang Sun
Abstract
This paper introduces an efficient collocation solver, the generalized finite difference method (GFDM) combined with the recent-developed scale-dependent time stepping method (SD-TSM), to predict the anomalous diffusion behavior on surfaces governed by surface time-fractional diffusion equations. In the proposed solver, the GFDM is used in spatial discretization and SD-TSM is used in temporal discretization. Based on the moving least square theorem and Taylor series, the GFDM introduces the stencil selection algorithms to choose the stencil support of a certain node from the whole discretization nodes on the surface. It inherits the similar properties from the standard FDM and avoids the mesh generation, which is available particularly for high-dimensional irregular discretization nodes. The SD-TSM is a non-uniform temporal discretization method involving the idea of metric, which links the fractional derivative order with the non-uniform discretization strategy. Compared with the traditional time stepping methods, GFDM combined with SD-TSM deals well with the low accuracy in the early period. Numerical investigations are presented to demonstrate the efficiency and accuracy of the proposed GFDM in conjunction with SD-TSM for solving either single or coupled fractional diffusion equations on surfaces.
A Appendix
A.1 Unit Sphere
The manifold is described implicitly as
The nodes distribution on unit sphere follows the minimum energy nodes set [37]. These nodes sets are distributed uniformly on the surface and the nodes in these sets are not oriented along any vertices or linesso it has nice property, which have been used successfully in many applications.
A.2 Red Blood Cell (RBC)
The Red Blood Cell is a mathematical model, first proposed by Evans et al [38], and its parametrical expression is
in which –π/2 ⩽ θ ⩽ π/2, – π ⩽ λ ⩽ π, r0 = 3.91/3.39, c0 = 0.81/3.39, c2 = 7.83/3.39, and c4 = –4.39/3.39.
A.3 Torus
The manifold is described implicitly by
The nodes distribution is similar with unit sphere by using that Reisz energy nodes set [39] (with a power of 2) is near minimal and it shares the similar good property.
A.4 Bumpy Sphere
The bumpy sphere is expressed parametrically as
where –π/2 ⩽ θ ⩽ π/2, 0 ⩽ λ ⩽ 2π.
Acknowledgements
The work described in this paper was supported by the National Science Fund of China (Grant Nos. 11772119, 11572111), Alexander von Humboldt Research Fellowship (ID: 1195938), the Foundation for Open Project of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics) (Grant No. MCMS-E-0519G01) and the Six Talent Peaks Project in Jiangsu Province of China (Grant No. 2019-KTHY-009).
References
[1] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley (1993).Suche in Google Scholar
[2] C. Ionescu, A. Lopes, D. Copot, J.A.T. Machado, J.H.T. Bates, The role of fractional calculus in modeling biological phenomena: A review. Commun. Nonlinear Sci. Numer. Simul. 51 (2017), 141–159.10.1016/j.cnsns.2017.04.001Suche in Google Scholar
[3] G.C. Wu, D. Baleanu, Chaos synchronization of the discrete fractional logistic map. Signal Processing 102 (2015), 444–447.10.1016/j.sigpro.2014.10.018Suche in Google Scholar
[4] Q. Yang, D. Chen, T. Zhao, Y. Chen, Fractional calculus in image processing: A review. Fract. Calc. Appl. Anal. 19, No 5 (2016), 1222– 1249; DOI: 10.1515/fca-2016-0063; https://www.degruyter.com/journal/key/FCA/19/5/htmlSuche in Google Scholar
[5] O. Özgen, Physics-Based Animation Models Using Fractional Calculus. UC Merced (2013).Suche in Google Scholar
[6] S.J. Ruuth, B. Merriman, A simple embedding method for solving partial differential equations on surfaces. J. Comput. Phys. 227, No 3 (2008), 1943–1961.10.1016/j.jcp.2007.10.009Suche in Google Scholar
[7] D. Qiang, M.D. Gunzburger, L. Ju, Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere. Comput. Meth. Appl. Mech. Eng. 192 (2003), 3933–3957.10.1016/S0045-7825(03)00394-3Suche in Google Scholar
[8] D. Calhoun, C. Helzel, A finite volume method for solving parabolic equations on logically cartesian curved surface meshes. SIAM J. Sci. Comput. 31, No 6 (2009), 4066–4099.10.1137/08073322XSuche in Google Scholar
[9] G. Dziuk, C.M. Elliott, Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, No 2 (2007), 262–292.10.1093/imanum/drl023Suche in Google Scholar
[10] E.J. Fuselier, G.B. Wright, A high-order kernel method for diffusion and reaction-diffusion equations on surfaces. J. Sci. Comput. 56 (2013), 535–565.10.1007/s10915-013-9688-xSuche in Google Scholar
[11] A. Petras, L. Ling, S.J. Ruuth, An RBF-FD closest point method for solving PDEs on surfaces. J. Comput. Phys. 370 (2018), 43–57.10.1016/j.jcp.2018.05.022Suche in Google Scholar
[12] H.G.Sun, W. Chen, Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling. Physica A. 388, No 21 (2009), 4586–4592.10.1016/j.physa.2009.07.024Suche in Google Scholar
[13] W. Chen, H.G. Sun, X.Zhang, D. Korošak, Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 59, No 5 (2010), 1754–1758.10.1016/j.camwa.2009.08.020Suche in Google Scholar
[14] H.G. Sun, A. Chang, Y. Zhang, W. Chen, A review on variable-order fractional differential equations: Mathematical foundations, physical models, and its applications. Fract. Calc. Appl. Anal. 22, No 1 (2018), 27–59; DOI: 10.1515/fca-2019-0003; https://www.degruyter.com/journal/key/FCA/22/1/htmlSuche in Google Scholar
[15] F. Liu, M. Meerschaert, R. McGough, P. Zhuang, Q.Liu, Numerical methods for solving the multiterm time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 9–25; DOI: 10.2478/s13540-013-0002-2; https://www.degruyter.com/journal/key/FCA/16/1/htmlSuche in Google Scholar PubMed PubMed Central
[16] A.H. Bhrawy, E.H. Doha, D. Baleanu, S.S. Ezz-Eldieng, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. 293 (2015), 142–156.10.1016/j.jcp.2014.03.039Suche in Google Scholar
[17] Y. Zhang, H.G. Sun, C. Zheng, Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application. Fract. Calc. Appl. Anal. 22, No 6 (2019), 1607–1640; DOI: 10.1515/fca-2019-0083; https://www.degruyter.com/journal/key/FCA/22/6/htmlSuche in Google Scholar
[18] C. Li, Z. Zhao, Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Com-put. Math. Appl. 62, No 3 (2011), 855–875.10.1016/j.camwa.2011.02.045Suche in Google Scholar
[19] M. Nickaeen, A. Ouazzi, S. Turek, Newton multigrid least-squares FEM for the VVP formulation of the Navier–Stokes equations. J. Com-put. Phys. 256 (2014), 416–427.10.1016/j.jcp.2013.09.011Suche in Google Scholar
[20] Z.J. Fu, W. Chen, H.T. Yang, Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. Phys. 235 (2013), 52–66.10.1016/j.jcp.2012.10.018Suche in Google Scholar
[21] M. Abbaszadeh, M. Dehghan, An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer. Algorithms. 75 (2016), 173–211.10.1007/s11075-016-0201-0Suche in Google Scholar
[22] K.C. Cheung, L. Ling, A kernel-based embedding method and convergence analysis for surfaces PDEs. SIAM J. Sci. Comput. 40, No 1 (2018), A266–A287.10.1137/16M1080410Suche in Google Scholar
[23] E.F. Bollig, N. Flyer, G. Erlebacher, Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs. J. Comput. Phys. 231 (2012), 7133–7151.10.1016/j.jcp.2012.06.030Suche in Google Scholar
[24] V. Shankar, G.B. Wright, R.M. Kirby, A.L. Fogelson, A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction–diffusion equations on surfaces. J. Sci. Comput. 63 (2015), 745– 768.10.1007/s10915-014-9914-1Suche in Google Scholar PubMed PubMed Central
[25] P. Suchde, J. Kuhnert, A Meshfree Generalized Finite Difference Method for Surface PDEs. Comput. Math. Appl. 78, No 8 (2019), 2789– 2805.10.1016/j.camwa.2019.04.030Suche in Google Scholar
[26] Z.J. Fu, Z.C. Tang, H.T. Zhao, P.W. Li, T. Rabczuk, Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method. Eur. Phys. J. Plus. 134 (2019), ID # 272.10.1140/epjp/i2019-12786-7Suche in Google Scholar
[27] Z.J. Fu, Z.Y. Xie, S.Y. Ji, C.C. Tsai, Meshless generalized finite difference method for water wave interactions with multiple-bottom-seated-cylinder-array structures. Ocean Eng. 195 (2020), ID # 106736.10.1016/j.oceaneng.2019.106736Suche in Google Scholar
[28] F.U. Prieto, J.J.B. Muñoz, L.G. Corvinos, Application of the generalized finite difference method to solve the advection–diffusion equation. J. Comput. Appl. Math. 235, No 7 (2011), 1849–1855.10.1016/j.cam.2010.05.026Suche in Google Scholar
[29] B. Froehle, P.O. Persson, A high-order discontinuous Galerkin method for fluid–structure interaction with efficient implicit–explicit time stepping. J. Comput. Phys. 272 (2014), 455–470.10.1016/j.jcp.2014.03.034Suche in Google Scholar
[30] J.S. Pang, V. Kumar, P. Song, Convergence of Time-Stepping Method for Initial and Boundary-Value Frictional Compliant Contact Problems. SIAM J. Numer. Anal. 43, No 5 (2005), 2200–2226.10.1137/040612269Suche in Google Scholar
[31] X.T. Liu, H.G. Sun, Y. Zhang, Z.J. Fu, A scale-dependent finite difference approximation for time fractional differential equation. Comput. Mech. 63 (2019), 429–442.10.1007/s00466-018-1601-xSuche in Google Scholar
[32] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Suche in Google Scholar
[33] M. Stynes, E. O’Riordan, J.L. Gracia Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation. SIAM J. Numer. Anal. 55, No 2 (2017), 1057–1079.10.1137/16M1082329Suche in Google Scholar
[34] Y.N. Zhang, Z.Z. Sun, H.L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265 (2014), 195–210.10.1016/j.jcp.2014.02.008Suche in Google Scholar
[35] C. Li, Q. Yi, A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316 (2016), 614–631.10.1016/j.jcp.2016.04.039Suche in Google Scholar
[36] B. Jin, R. Lazarov, Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview. Comput. Meth. Appl. Mech. Eng. 346 (2018), 332–358.10.1016/j.cma.2018.12.011Suche in Google Scholar
[37] R.S. Womersley, I.H. Sloan, Interpolation and cubature on the sphere. http://web.maths.unsw.edu.au/rsw/Sphere/ (2013).Suche in Google Scholar
[38] E. Evans, Y.C. Fung, Improved measurements of the erythrocyte geometry. Microvasc. Res. 4, No 4 (1972), 335–347.10.1016/0026-2862(72)90069-6Suche in Google Scholar
[39] D. Hardin, E. Saff, Discretizing manifolds via minimum energy points. Notices Am. Math. Soc. 51, No 10 (2004), 118–1194.Suche in Google Scholar
© 2021 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial Survey
- In memory of the honorary founding editors behind the FCAA success story
- Research Paper
- Short time coupled fractional fourier transform and the uncertainty principle
- (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response
- Sharp asymptotics in a fractional Sturm-Liouville problem
- Multi-term fractional integro-differential equations in power growth function spaces
- Galerkin method for time fractional semilinear equations
- Müntz sturm-liouville problems: Theory and numerical experiments
- Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u
- Nonlinear convolution integro-differential equation with variable coefficient
- An efficient localized collocation solver for anomalous diffusion on surfaces
- Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach
- Sliding methods for the higher order fractional laplacians
- Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
Artikel in diesem Heft
- Frontmatter
- Editorial Survey
- In memory of the honorary founding editors behind the FCAA success story
- Research Paper
- Short time coupled fractional fourier transform and the uncertainty principle
- (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response
- Sharp asymptotics in a fractional Sturm-Liouville problem
- Multi-term fractional integro-differential equations in power growth function spaces
- Galerkin method for time fractional semilinear equations
- Müntz sturm-liouville problems: Theory and numerical experiments
- Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u
- Nonlinear convolution integro-differential equation with variable coefficient
- An efficient localized collocation solver for anomalous diffusion on surfaces
- Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach
- Sliding methods for the higher order fractional laplacians
- Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
