Discrete Dynamics in Nature and Society

Volume 2015, Article ID 303857, 9 pages

http://dx.doi.org/10.1155/2015/303857

## A Discrete Method Based on the CE-SE Formulation for the Fractional Advection-Dispersion Equation

Department of Applied Mathematics, CIMAT, Jalisco s/n, 36240 Guanajuato, GTO, Mexico

Received 16 November 2014; Revised 15 January 2015; Accepted 17 January 2015

Academic Editor: Manuel De la Sen

Copyright © 2015 Silvia Jerez and Ivan Dzib. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We obtain a numerical algorithm by using the space-time conservation element and solution element (CE-SE) method for the fractional advection-dispersion equation. The fractional derivative is defined by the Riemann-Liouville formula. We prove that the CE-SE approximation is conditionally stable under mild requirements. A numerical simulation is performed for the one-dimensional case by considering a benchmark with a discontinuous initial condition in order to compare the results with the analytical solution.

#### 1. Introduction

Recently, the analytical and numerical framework of fractional differential equations has attracted much attention because of its extended use in the modeling of nonlocal transport phenomena which appear in many fields like magnetic plasma, turbulence, and flows in porous media [1–5]. A nonlocal transport can lead an anomalous diffusion because of large tracer displacements (super-diffusion) or trapping structure vortices (sub-diffusion). The standard local diffusion description is linked to Gaussian processes and diffusion differential models are obtained. However, a nonlocal transport flow presents a non-Gaussian particle motion and in this case fractional diffusion models are derived [6–8]. One of the most considered proposals for the modeling of the mass flow in media with anomalous diffusion is the fractional advection-dispersion equation (FADE); see, for example, [7–10].

Different definitions for the fractional derivative can be used to describe the FADE, and the most common ones are the Caputo, Riemann-Liouville, and Grünwald-Letnikov derivatives [11–13]. Here, we consider the one-dimensional FADE given by and use the Riemann-Liouville formula to describe the fractional derivatives: where and represent the space and the time, respectively, is the solute concentration, is the fractional dispersion, is the skewness of the diffusive transport taking values in the interval , and is the fractional order. In our case we consider and by definition of the ceil function we get . We can rewrite the fractional equation (1) in a more compact form as follows: by defining the fractional operator as

The analytical solution of (3) is only known under particular initial and boundary conditions, so numerical solutions are calculated and studied for most of the problems. In the last years, a lot of works have been focused on the development of numerical algorithms for fractional differential equations. The first efforts are based on extensions of standard methods used for the nonfractional case like the finite difference method (FDM) [14–17], the finite volume method (FVM) [18, 19], and the finite element method (FEM) [20, 21]. More recently other methods have been developed, for example, the spectral method [22], the collocation method [23], or a combination of both [24]. Generally, the FDM is constructed using the shifted Grünwald formula for the discretization of the fractional derivative, the FVM under a conservative formulation is based on the discretization of the integral using the Riemann-Liouville definition or the Caputo definition of the fractional derivative, and the FEM and collocation methods are variational techniques based on the discretization of the integral on some interpolating polynomial that approximates the solution. Therefore, all these techniques generate algorithms with dense coefficient matrices for which it is complicated to analyze their stability and a high computational cost to solve them is required.

As an alternative to avoid the discretization of the integral associated with the fractional derivative, in this paper we propose a numerical algorithm based on the space-time conservation element and solution element (CE-SE) method developed by Chang and To [25]. For this method, the solution is approximated by a first-order Taylor expansion which must fulfill the integral form of the equation in the conservation elements and the differential form in the solution elements. Therefore, the space-time domain is discretized two times by: a rhomboid mesh and a rectangular mesh. The theoretical framework of the CE-SE method for the nonfractional case is well established and it has been applied to a large number of real problems with successful results [26–33]. An important property of the fractional differential operators is the nonlocality which is achieved with the discretization of the integral associated with the fractional derivative. In our case we seek to keep the nonlocality by providing information of the numerical solution at nearby points in both meshes and using nonlocal approximations of the standard derivatives. It is important to remark that we consider this work as a first step to open a venue for a different way to approximate the fractional differential equations like the FADE equation.

The paper is organized as follows. In Section 2, we give the discretization of the space-time domain for the CE-SE method, its extension to the fractional advection-dispersion equation (4), and the von Neumann stability analysis of the proposed algorithm. In Section 3, we present numerical experiments of the fractional CE-SE method for the FADE with an initial condition type shock and it is compared with the analytical solution of the problem. Finally, this work is completed with a section of conclusions and an appendix with the calculation of the fractional Riemann-Liouville of a linear polynomial of two variables.

#### 2. CE-SE Method for the 1D FADE

##### 2.1. Discretization of the Space-Time Domain

An important feature of the CE-SE method is how the space-time domain is discretized. Let us start with the usual discretization by taking a set of points such that for and for each we take such that with being the left end of the spatial interval and the spatial and time step sizes are defined by and with being a constant of proportionality. Next, the space-time domain is partitioned by two types of mesh: nonoverlapping opened rhombi centered in the solution node which are named solution elements (SEs) and nonoverlapping closed rectangles which are called conservation element (CEs); see Figure 1.