# zbMATH — the first resource for mathematics

Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. (English) Zbl 07235959
Summary: In this paper, we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. The main challenges of the numerical schemes come from the singularity in the time direction. When $$0 < H < 0.5$$, a change of variables $$\partial (t^{2H} )=2Ht^{2H-1}\partial t$$ avoids the singularity of numerical computation at $$t = 0$$, which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For $$H > 0.5$$, the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational efficiency. The stability and convergence of the numerical schemes are demonstrated by using Fourier method. By numerically solving the corresponding Fokker-Planck equation, we obtain the mean squared displacement of stochastic processes, which conforms to the characteristics of the tempered fractional Brownian motion.
##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 60J65 Brownian motion 60G22 Fractional processes, including fractional Brownian motion 35Q84 Fokker-Planck equations
FODE
Full Text:
##### References:
 [1] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1, 1-77 (2000) · Zbl 0984.82032 [2] Bruno, R.; Sorriso-Valvo, L.; Carbone, V.; Bavassano, B., A possible truncated-Lévy-flight statistics recovered from interplanetary solar-wind velocity and magnetic-field fluctuations, Europhys. Lett., 66, 1, 146-152 (2004) [3] Chen, Y.; Wang, XD; Deng, WH, Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion, J. Stat. Phys., 169, 1, 18-37 (2017) · Zbl 1397.82042 [4] Deng, WH; Zhang, ZJ, High Accuracy Algorithm for the Differential Equations Governing Anomalous Diffusion (2019), Singapore: World Scientific, Singapore [5] Deng, WH; Li, BY; Tian, WY; Zhang, PW, Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian, Multiscale Model Simul., 16, 1, 125-149 (2018) [6] Deng, WH; Wu, XC; Wang, WL, Mean exit time and escape probability for the anomalous processes with the tempered power-law waiting times, EPL., 117, 1, 10009 (2017) [7] Drysdale, PM; Robinson, PA, Lévy random walks in finite systems, Phys. Rev. E., 58, 5, 5382-5394 (1998) [8] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A., 37, 31, R161-R208 (2004) · Zbl 1075.82018 [9] Schumer, R.; Meerschaert, MM; Baeumer, B., Fractional advection-dispersion equations for modeling transport at the Earth surface, J. Geophys. Res., 114, 6, F00A07 (2009) [10] Chen, Y.; Wang, XD; Deng, WH, Tempered fractional Langevin-Brownian motion with inverse β-stable subordinator, J. Phys. A: Math. Theor., 51, 49, 495001 (2018) · Zbl 1411.82031 [11] Meerschaert, MM; Sabzikar, F., Tempered fractional Brownian motion, Stat. Probab. Lett., 83, 10, 2269-2275 (2013) · Zbl 1287.60050 [12] Meerschaert, MM; Sabzikar, F., Stochastic integration for tempered fractional Brownian motion, Stoch. Process. Appl., 124, 7, 2363-2387 (2014) · Zbl 1329.60166 [13] Yuste, SB; Quintana-Murillo, J., A finite difference method with non-uniform timesteps for fractional diffusion equations, Comput. Phys. Comm., 183, 12, 2594-2600 (2012) · Zbl 1268.65120 [14] Filbet, F.; Pareschi, L., A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the nonhomogeneous case, J. Comput. Phys., 179, 1, 1-26 (2002) · Zbl 1003.82011 [15] Zhang, YN; Sun, ZZ, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230, 24, 8713-8728 (2011) · Zbl 1242.65174 [16] Jin, S.; Yan, B., A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation, J. Comput. Phys., 230, 17, 6420-6437 (2011) · Zbl 1408.76594 [17] Zhuang, P.; Liu, F., Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput., 22, 3, 87-99 (2006) · Zbl 1140.65094 [18] Chen, CM; Liu, F.; Turner, I.; Anh, V., Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algor., 54, 1, 1-21 (2010) · Zbl 1191.65116 [19] Smith, J., Engineering applications of ADI methods to piecewise linear multidimensional heat transfer, J. Comput. Phys., 17, 2, 181-191 (1975) · Zbl 0293.65090 [20] Epperlein, EM, Implicit and conservative difference scheme for the Fokker-Planck equation, J. Comput. Phys., 112, 2, 291-297 (1994) · Zbl 0806.76050 [21] Reichmann, O., Optimal space-time adaptive wavelet methods for degenerate parabolic PDEs, Numer. Math., 121, 2, 337-365 (2012) · Zbl 1250.65118 [22] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 46, 2, 1079-1095 (2008) · Zbl 1173.26006 [23] Napolitano, M., Efficient ADI and spline ADI methods for the steady-state Navier-Stokes equations, Int. J. Numer. Meth. Fluids., 4, 12, 1101-1115 (1984) · Zbl 0596.76033 [24] Yuste, SB; Acedo, L., An explicit finite difference method and a new von neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42, 5, 1862-1874 (2005) · Zbl 1119.65379 [25] Lemou, M.; Mieussens, L., Implicit schemes for the Fokker-Planck-Landau equation, SIAM J. Numer. Anal., 27, 3, 809-830 (2005) · Zbl 1096.82015 [26] Zhang, YN; Sun, ZZ, Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comput., 59, 1, 104-128 (2014) · Zbl 1304.65208 [27] Shen, S.; Liu, F., Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends, ANZIAM J., 46, E, 871-887 (2005) [28] Murio, DA, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56, 4, 1138-1145 (2008) · Zbl 1155.65372 [29] Langlands, TAM; Henry, BI, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205, 2, 719-736 (2005) · Zbl 1072.65123
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.