An adaptive non-uniform L2 discretization for the one-dimensional space-fractional Gray-Scott system. (English) Zbl 07899939
Summary: This paper introduces a new numerical method for solving space-fractional partial differential equations (PDEs) on non-uniform adaptive finite difference meshes, considering a fractional order \(\alpha\in(1,2)\) in one dimension. The fractional Laplacian in PDE is computed by using Riemann-Liouville (R-L) derivatives, incorporating a boundary condition of the form \(u=0\) in \(\mathbb{R}\backslash\Omega\). The proposed approach extends the L2 method to non-uniform meshes for calculating the R-L derivatives. The spatial mesh generation employs adaptive moving finite differences, offering adaptability at each time step through grid reallocation based on previously calculated solutions. The chosen mesh movement technique, moving mesh PDE-5 (MMPDE-5), demonstrates rapid and efficient mesh movement. The numerical solutions are obtained by applying the non-uniform L2 numerical scheme and the MMPDE-5 method for moving meshes automatically. Two numerical experiments focused on the space-fractional heat equation validate the convergence of the proposed scheme. The study concludes by exploring patterns in equations involving the fractional Laplacian term within the Gray-Scott system. It reveals self-replication, travelling wave, and chaotic patterns, along with two distinct evolution processes depending on the order \(\alpha\): from self-replication to standing waves and from travelling waves to self-replication.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 26Axx | Functions of one variable |
| 35Rxx | Miscellaneous topics in partial differential equations |
References:
| [1] | Failla, G.; Zingales, M., Advanced materials modelling via fractional calculus: challenges and perspectives, Phil Trans R Soc A, 378, 2172, Article 20200050 pp., 2020 |
| [2] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, 2006, Elsevier · Zbl 1092.45003 |
| [3] | Oldham, K.; Spanier, J., The fractional calculus theory and applications of differentiation and integration to arbitrary order, 1974, Elsevier Science · Zbl 0292.26011 |
| [4] | Gorenflo, R.; Mainardi, F., Random walk models approximating symmetric space-fractional diffusion processes, (Problems and methods in mathematical physics: the Siegfried Prössdorf memorial volume proceedings of the 11th TMP, 2001, Springer), 120-145 · Zbl 1007.60082 |
| [5] | Lischke, A.; Pang, G.; Gulian, M.; Song, F.; Glusa, C.; Zheng, X., What is the fractional Laplacian? A comparative review with new results, J Comput Phys, 404, Article 109009 pp., 2020 · Zbl 1453.35179 |
| [6] | Bonito, A.; Borthagaray, J. P.; Nochetto, R. H.; Otárola, E.; Salgado, A. J., Numerical methods for fractional diffusion, Comput Vis Sci, 19, 5-6, 19-46, 2018 · Zbl 07704543 |
| [7] | Bogdan, K.; Burdzy, K.; Chen, Z.-Q., Censored stable processes, Probab Theory Related Fields, 127, 89-152, 2003 · Zbl 1032.60047 |
| [8] | Guan, Q.-Y.; Ma, Z.-M., Boundary problems for fractional Laplacians, Stoch Dyn, 5, 03, 385-424, 2005 · Zbl 1077.60036 |
| [9] | Yang, Q.; Liu, F.; Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl Math Model, 34, 1, 200-218, 2010 · Zbl 1185.65200 |
| [10] | Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J Comput Appl Math, 166, 1, 209-219, 2004 · Zbl 1036.82019 |
| [11] | Li, C.; Zeng, F., Numerical methods for fractional calculus, vol. 24, 2015, CRC Press · Zbl 1326.65033 |
| [12] | Lyu, P.; Vong, S., A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin-Bona-Mahony-type equation with nonsmooth solutions, Numer Methods Partial Differential Equations, 36, 3, 579-600, 2020 · Zbl 07771404 |
| [13] | Alikhanov, A. A.; Huang, C., A high-order L2 type difference scheme for the time-fractional diffusion equation, Appl Math Comput, 411, Article 126545 pp., 2021 · Zbl 1510.65184 |
| [14] | Huang, W.; Ren, Y.; Russell, R. D., Moving mesh partial differential equations (MMPDEs) based on the equidistribution principle, SIAM J Numer Anal, 31, 3, 709-730, 1994 · Zbl 0806.65092 |
| [15] | Huang, W.; Russell, R. D., Adaptive moving mesh methods, 2010, Springer Science & Business Media |
| [16] | Dorfi, E.; Drury, L., Simple adaptive grids for 1-D initial value problems, J Comput Phys, 69, 1, 175-195, 1987 · Zbl 0607.76041 |
| [17] | Huang, W.; Russell, R. D., Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J Sci Comput, 20, 3, 998-1015, 1998 · Zbl 0956.76076 |
| [18] | Huang, W., Practical aspects of formulation and solution of moving mesh partial differential equations, J Comput Phys, 171, 2, 753-775, 2001 · Zbl 0990.65107 |
| [19] | Lee, K.-J.; McCormick, W. D.; Pearson, J. E.; Swinney, H. L., Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369, 6477, 215-218, 1994 |
| [20] | Pearson, J. E., Complex patterns in a simple system, Science, 261, 5118, 189-192, 1993 |
| [21] | Doelman, A.; Kaper, T. J.; Zegeling, P. A., Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10, 2, 523, 1997 · Zbl 0905.35044 |
| [22] | Reynolds, W. N.; Pearson, J. E.; Ponce-Dawson, S., Dynamics of self-replicating patterns in reaction diffusion systems, Phys Rev Lett, 72, 17, 2797, 1994 |
| [23] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: Theory and applications, 1993 · Zbl 0818.26003 |
| [24] | Stinga, P. R.; Torrea, J. L., Extension problem and Harnack’s inequality for some fractional operators, Comm Partial Differential Equations, 35, 11, 2092-2122, 2010 · Zbl 1209.26013 |
| [25] | Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull Sci Math, 136, 5, 521-573, 2012 · Zbl 1252.46023 |
| [26] | King, F. W., Hilbert transforms: Volume 1, Encyclopedia of mathematics and its applications, 2009, Cambridge University Press · Zbl 1188.44005 |
| [27] | Cayama, J.; Cuesta, C. M.; de la Hoz, F., A pseudospectral method for the one-dimensional fractional Laplacian on \(\mathbb{R} \), Appl Math Comput, 389, Article 125577 pp., 2021 · Zbl 1508.65090 |
| [28] | De Boor, C., Good approximation by splines with variable knots. II, (Conference on the numerical solution of differential equations: dundee 1973, 2006, Springer), 12-20 · Zbl 0343.65005 |
| [29] | Huang, W.; Ren, Y.; Russell, R. D., Moving mesh methods based on moving mesh partial differential equations, J Comput Phys, 113, 2, 279-290, 1994 · Zbl 0807.65101 |
| [30] | Huang, W.; Sun, W., Variational mesh adaptation II: error estimates and monitor functions, J Comput Phys, 184, 2, 619-648, 2003 · Zbl 1018.65140 |
| [31] | Furzeland, R.; Verwer, J.; Zegeling, P., A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines, J Comput Phys, 89, 2, 349-388, 1990 · Zbl 0705.65066 |
| [32] | Shampine, L. F., Solving \(0 = F ( t , y ( t ) , y^\prime ( t ) )\) in Matlab, J Numer Math, 10, 4, 291-310, 2002 · Zbl 1019.65044 |
| [33] | Franz, S.; Kopteva, N., Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations, J Comput Appl Math, 427, Article 115122 pp., 2023 · Zbl 1514.65116 |
| [34] | Kautsky, J.; Nichols, N., Equidistributing meshes with constraints, SIAM J Sci Stat Comput, 1, 4, 499-511, 1980 · Zbl 0455.65068 |
| [35] | Huang, W.; Russell, R. D., Analysis of moving mesh partial differential equations with spatial smoothing, SIAM J Numer Anal, 34, 3, 1106-1126, 1997 · Zbl 0874.65071 |
| [36] | Srivastava, H.; Agarwal, P.; Jain, S., Generating functions for the generalized Gauss hypergeometric functions, Appl Math Comput, 247, 348-352, 2014 · Zbl 1338.33015 |
| [37] | Duo, S.; van Wyk, H. W.; Zhang, Y., A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J Comput Phys, 355, 233-252, 2018 · Zbl 1380.65323 |
| [38] | lomiej Dyda, B., Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract Calc Appl Anal, 15, 4, 536-555, 2012 · Zbl 1312.35176 |
| [39] | Gutleb, T. S.; Papadopoulos, I. P., Explicit fractional Laplacians and Riesz potentials of classical functions, 2023, arXiv preprint arXiv:2311.10896 |
| [40] | Wang, T.; Song, F.; Wang, H.; Karniadakis, G. E., Fractional Gray-Scott model: well-posedness, discretization, and simulations, Comput Methods Appl Mech Engrg, 347, 1030-1049, 2019 · Zbl 1440.35344 |
| [41] | Nishiura, Y.; Ueyama, D., Spatio-temporal chaos for the Gray-Scott model, Physica D, 150, 3-4, 137-162, 2001 · Zbl 0981.35022 |
| [42] | Kaslik, E.; Sivasundaram, S., Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions, Nonlinear Anal Real World Appl, 13, 3, 1489-1497, 2012 · Zbl 1239.44001 |
| [43] | Area, I.; Losada, J.; Nieto, J. J., On fractional derivatives and primitives of periodic functions, (Abstract and applied analysis, vol. 2014, 2014, Hindawi) · Zbl 1470.26008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
