Ercília Sousa

Finite difference approximations for a fractional advection diffusion problem

The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one-dimensional advection-diffusion model, where the usual second-order derivative gives place to a fractional derivative of order @a, with 1<@a=<2. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.

A second order explicit finite difference method for the fractional advection diffusion equation

We develop a numerical method for fractional advection diffusion problems with source terms in domains with homogeneous boundary conditions. The numerical method is derived by using a Lax-Wendroff-type time discretization procedure, it is explicit and second order accurate. The convergence of the numerical method is studied and numerical results are presented.

HOW TO APPROXIMATE THE FRACTIONAL DERIVATIVE OF ORDER 1 <α ≤ 2

The fractional derivative of order α, with 1 < α ≤ 2 appears in several diffusion problems used in physical and engineering applications. Therefore to obtain highly accurate approximations for this derivative is of great importance. Here, we describe and compare different numerical approximations for the fractional derivative of order 1 < α ≤ 2. These approximations arise mainly from the Grunwald–Letnikov definition and the Caputo definition and they are consistent of order one and two. In the end some numerical examples are given, to compare their performance.

The controversial stability analysis

In this review we present different techniques for obtaining stability limits for a finite difference scheme--the forward-time and space-centered numerical scheme applied to the convection-diffusion equation. A survey of past attempts to state stability conditions for this scheme illustrates the difficulties in stability analysis that arise as soon as a scheme becomes more complex and illuminates the concepts of necessary and sufficient conditions for stability.

An explicit high order method for fractional advection diffusion equations

We propose a high order explicit finite difference method for fractional advection diffusion equations. These equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order α with 1 < α ? 2 . This operator is defined by a combination of the left and right Riemann-Liouville fractional derivatives. We study the convergence of the numerical method through consistency and stability. The order of convergence varies between two and three and for advection dominated flows is close to three. Although the method is conditionally stable, the restrictions allow wide stability regions. The analysis is confirmed by numerical examples.

On the influence of numerical boundary conditions

Our understanding about the behaviour of numerical solutions for evolutionary convection-diffusion equations is mainly based on analysis of infinite domains situations with stability given by yon Neumann analysis. Almost all practical problems involve physical domains with boundaries. For evolution problems with Dirichlet boundary conditions, some algorithms can be used without alteration near a boundary. However, the application of higher order methods such as Quickest or second order upwinding introduces difficulty near an inflow boundary, since for interior points adjacent to the boundary there are insufficient upstream points for the high order scheme to be applied without alteration. For that reason such methods require a careful treatment on the inflow boundary, where additional numerical boundary conditions have to be introduced. The choice of numerical boundary conditions turns out to be crucial for stability. A test problem is described, showing the practical advantages of some numerical boundary conditions versus the others by comparison with an exact solution.

Numerical Solution for a Non-Fickian Diffusion in a Periodic Potential

Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and meansquare-displacement (covering both inertial and diffusive regimes) are presented.

An alternating direction implicit method for a second-order hyperbolic diffusion equation with convection

A numerical method is presented to solve a two-dimensional hyperbolic diffusion problem where is assumed that both convection and diffusion are responsible for flow motion. Since direct solutions based on implicit schemes for multidimensional problems are computationally inefficient, we apply an alternating direction method which is second order accurate in time and space. The stability of the alternating direction method is analyzed using the energy method. Numerical results are presented to illustrate the performance in different cases.

Stability Analysis of Difference Methods for Parabolic Initial Value Problems

A decomposition of the numerical solution can be defined by the normal mode representation, that generalizes further the spatial eigenmode decomposition of the von Neumann analysis by taking into account the boundary conditions which are not periodic. In this paper we present some new theoretical results on normal mode analysis for a linear and parabolic initial value problem. Furthermore we suggest an algorithm for the calculation of stability regions based on the normal mode theory.

A Godunov-Ryabenkii Instability for a Quickest Scheme

We consider a finite difference scheme, called Quickest, introduced by Leonard in 1979, for the convection-diffusion equation. Quickest uses an explicit, Leith-type differencing and third-order upwinding on the convective derivatives yielding a four-point scheme. For that reason the method requires careful treatment on the inflow boundary considering the fact that we need to introduce numerical boundary conditions and that they could lead us to instability phenomena. The stability region is found with the help of one of the most powerful methods for local analysis of the influence of boundary conditions -the Godunov-Ryabenkii theory.