Journal of Computational and Applied Mathematics 233 (2010) 2438–2448 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Tempered stable Lévy motion and transient super-diffusion Boris Baeumer a,∗ , Mark M. Meerschaert b a Department of Mathematics & Statistics, University of Otago, Dunedin, New Zealand b Department of Statistics & Probability, Michigan State University, Wells Hall, E. Lansing, MI 48824, United States article info Article history: Received 23 September 2008 Received in revised form 8 October 2009 Keywords: Fractional derivatives Particle tracking Power law Truncated power law abstract The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power- law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank–Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes. © 2009 Elsevier B.V. All rights reserved. 1. Introduction The diffusion equation ∂ t p = c ∂ 2 x p governs the transition densities of a Brownian motion B(t ), and its solutions spread at the rate t 1/2 for all time. The space-fractional diffusion equation ∂ t p = c ∂ α x p for 0 <α< 2 governs the transition densities of a totally skewed α-stable Lévy motion S (t ), and its solutions spread at the super-diffusive rate S (ct ) ∼ c 1/α S (t ) for any time scale c . The super-diffusive spreading is the result of large power-law jumps, with the probability of a jump longer than r falling off like r −α [1–3]. The super-diffusive scaling of a stable Lévy motion cannot be expressed in terms of the second moment, which fails to exist due to the power-law tail of the probability density [4]. Instead one can use fractional moments [5] or quantiles. The totally skewed α-stable Lévy motion S (t ) is the scaling limit of a random walk with power-law jumps: c −1/α (X 1 + ···+ X [ct ] ) ⇒ S (t ) in distribution as the time scale c →∞, when the independent jumps all satisfy P (X > r ) = Ar −α for large x > 0. Here 0 <α< 2 so that the second moment is infinite, and the usual Gaussian central limit theorem does not apply. The random walk with power-law jumps is called a Lévy flight [6]. The order of the fractional derivative in the governing equation ∂ t p = c ∂ α x p equals the power-law index of the jumps. A more general jump variable with P (X < −r ) = qAr −α and P (X > r ) = (1 − q)Ar −α leads to a governing equation ∂ t p = cq∂ α −x p + (1 − q)c ∂ α x p with 0 ≤ q ≤ 1. A continuous time random walk (CTRW) imposes a random waiting time T n before the jumps X n , and power-law waiting times P (T > t ) = Bt −β for 0 <β< 1 lead to a space–time fractional governing equation ∂ β t p = c ∂ α x p. See [7] for more details. Fractional diffusion equations are important in applications to physics [2,3], finance [8], and hydrology [9]. Truncated Lévy flights were proposed by Mantegna and Stanley [10,11] as a modification of the α-stable Lévy motion, since many deem the possibility of arbitrarily long jumps, and the mathematical fact of infinite moments, physically unrealistic. In that model, the largest jumps are simply discarded. Some other modifications to achieve finite second moments were proposed by Sokolov et al. [12], who add a higher-order power-law factor, and Chechkin et al. [13], who ∗ Corresponding author. E-mail addresses: bbaeumer@maths.otago.ac.nz (B. Baeumer), mcubed@stt.msu.edu (M.M. Meerschaert). 0377-0427/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2009.10.027