Past Event: Oden Institute Seminar

Nonlocal transport modeling: fundamentals, applications, and numerical methods (3rd of four 1.5 hour lectures) - NOTE: start time change

Diego de Castillo-Negrete, Senior Research Scientist, Oak Ridge National Laboratory; and Faculty member, University of Tennessee

Abstract

Transport modeling is a key element of applied sciences and engineering, and a fertile area of applied mathematics. In the past, a significant amount of work has been devoted to models formulated in terms of local partial differential equations (e.g., linear and nonlinear advection-diffusion type equations). However, relatively recently there has been growing experimental and theoretical evidence that these models fail to describe anomalous non-diffusive transport (e.g., super-diffusive and sub-diffusive transport). To describe these phenomena, nonlocal models introduce nonlocal flux-gradient relations and formulate transport in terms of partial integro-differential equations. The goal of these lectures is to present an introduction to the state of art of nonlocal transport modeling with emphasis on basic mathematical aspects and practical numerical methods. Applications will play a prominent role throughout the lectures in motivating the need for nonlocality, and in guiding the mathematical and numerical developments. Although the main focus will be on fractional diffusion type models, we will devote time to other nonlocal descriptions. Following an overview of nonlocal transport, the first part of the lectures will cover the fundamentals including the statistical foundation of nonlocal models (based on the theory of continuous time random walks driven by general Levy processes) and the basic mathematical properties of fractional derivatives including the fundamental solutions of fractional diffusion equations. Both, spatial nonlocality (resulting, e.g. from large non-Gaussian jumps) and temporal nonlocality (resulting, e.g. from non-Markovian memory effects) will be considered in one and higher dimensions. The second part of the lectures will focus on practical numerical algorithms based on finite-difference continuum methods and stochastic Langevin particle-based methods. A careful discussion of the nontrivial role played by boundary conditions when solving nonlocal models in finite-size domains will be presented. Although applications will be mentioned throughout the first two parts of lectures, the third part will be devoted to an in-depth discussion of some specific applications including: (i) Nonlocal particle and heat transport in fluids and plasmas; (ii) Fluctuation-driven transport in the nonlocal Fokker-Planck equation (e.g., Levy ratchets); (iii) Nonlocal reaction diffusion systems (e.g., front acceleration in the nonlocal Fisher-Kolmogorov equation); (iv) Nonlocal models of option prices in markets with jumps based on the fractional Black-Scholes equation.