S. B. Yuste

An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations

A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Ficks law. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald-Letnikov definition of the fractional derivative operator to obtain an explicit fractional FTCS scheme for solving the fractional diffusion equation. The resulting method is amenable to a stability analysis a la von Neumann. We show that the analytical stability bounds are in excellent agreement with numerical tests. Comparison between exact analytical solutions and numerical predictions are made.A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald--Letnikov discretization of the Riemann--Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.

Reaction front in an A+B-->C reaction-subdiffusion process.

We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.

Equation of state of a multicomponent d-dimensional hard-sphere fluid

A simple recipe to derive the compressibility factor of a multicomponent mixture of d-dimensional additive hard spheres in terms of that of the one-component system is proposed. The recipe is based (i) on an exact condition that has to be satisfied in the special limit where one of the components corresponds to point particles; and (ii) on the form of the radial distribution functions at contact as obtained from the Percus—Yevick equation in the three-dimensional system. The proposal is examined for hard discs and hard spheres by comparison with well-known equations of state for these systems and with simulation data. In the special case of d = 3, our extension to mixtures of the Carnahan—Starling equation of state yields a better agreement with simulation than the already accurate Boublik—Mansoori—Carnahan—Starling—Leland equation of state.

An Explicit Difference Method for Solving Fractional Diffusion and Diffusion-Wave Equations in the Caputo Form

An explicit difference method is considered for solving fractional diffusion and fractional diffusion-wave equations where the time derivative is a fractional derivative in the Caputo form. For the fractional diffusion equation, the L1 discretization formula of the fractional derivative is employed, whereas the L2 discretization formula is used for the fractional diffusion-wave equation. In both equations, the spatial derivative is approximated by means of the three-point centered formula. The accuracy of the present method is similar to other well-known explicit difference schemes, but its region of stability is larger. The stability analysis is carried out by means of a kind of fractional von Neumann (or Fourier) method. The stability bound so obtained, which is given in terms of the Riemann zeta function, is checked numerically.

A finite difference method with non-uniform timesteps for fractional diffusion equations

Abstract An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the behavior of the solution in order to keep the numerical errors small without the penalty of a huge computational cost. The method is unconditionally stable and convergent. In fact, it is shown that consistency and stability implies convergence for a rather general class of fractional finite difference methods to which the present method belongs. The huge computational advantage of adaptive methods against fixed step methods for fractional diffusion equations is illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.

Pair correlation function of short-ranged square-well fluids.

We have performed extensive Monte Carlo simulations in the canonical (NVT) ensemble of the pair correlation function for square-well fluids with well widths lambda-1 ranging from 0.1 to 1.0, in units of the diameter sigma of the particles. For each one of these widths, several densities rho and temperatures T in the ranges 0.1< or =rhosigma(3)< or =0.8 and T(c)(lambda) less or approximately T less or approximately 3T(c)(lambda), where T(c)(lambda) is the critical temperature, have been considered. The simulation data are used to examine the performance of two analytical theories in predicting the structure of these fluids: the perturbation theory proposed by Tang and Lu [Y. Tang and B. C.-Y. Lu, J. Chem. Phys. 100, 3079 (1994); 100, 6665 (1994)] and the nonperturbative model proposed by two of us [S. B. Yuste and A. Santos, J. Chem. Phys. 101 2355 (1994)]. It is observed that both theories complement each other, as the latter theory works well for short ranges and/or moderate densities, while the former theory works for long ranges and high densities.

Contact values of the radial distribution functions of additive hard-sphere mixtures in d dimensions: A new proposal

The contact values gij(σij) of the radial distribution functions of a d-dimensional mixture of (additive) hard spheres are considered. A “universality” assumption is put forward, according to which gij(σij)=G(η,zij), where G is a common function for all the mixtures of the same dimensionality, regardless of the number of components, η is the packing fraction of the mixture, and zij=(σiσj/σij)〈σd−1〉/〈σd〉 is a dimensionless parameter, 〈σn〉 being the nth moment of the diameter distribution. For d=3, this universality assumption holds for the contact values of the Percus–Yevick approximation, the scaled particle theory, and, consequently, the Boublik–Grundke–Henderson–Lee–Levesque approximation. Known exact consistency conditions are used to express G(η,0), G(η,1), and G(η,2) in terms of the radial distribution at contact of the one-component system. Two specific proposals consistent with the above-mentioned conditions (a quadratic form and a rational form) are made for the z dependence of G(η,z). For one-dim...

On three explicit difference schemes for fractional diffusion and diffusion-wave equations

Three explicit difference schemes for solving fractional diffusion and fractional diffusion-wave equations are studied. We consider these equations in both the Riemann–Liouville and the Caputo forms. We find that the Gorenflo et al (2000 J. Comput. Appl. Math. 118 175) and the Yuste–Acedo (2005 SIAM J. Numer. Anal. 42 1862) methods when applied to fractional diffusion equations are equivalent when BDF1 coefficients are used to discretize the fractional derivative operators, but that this is not the case for fractional diffusion-wave equations. The accuracy and stability of the three methods are studied. Surprisingly, the third method, that of Ciesielski–Leszczynski (2003 Proc. 15th Conf. on Computer Methods in Mechanics), although closely related to the Gorenflo et al method, is the least accurate, especially for short times. The stability analysis is carried out by means of a procedure close to the standard von Neumann method. We find that the stability bounds of the three methods are the same.

Structure of multi-component hard-sphere mixtures

A method to obtain (approximate) analytical expressions for the radial distribution functions and structure factors in a multi-component mixture of additive hard spheres is introduced. In this method, only contact values of the radial distribution function and the isothermal compressibility are required and thermodynamic consistency is achieved. The approach is simpler than but yields equivalent results to the Generalized Mean Spherical Approximation. Calculations are presented for a binary and a ternary mixture at high density in which the BoublikMansoori-Carnahan-Starling-Leland equation of state is used. The results are compared with the Percus-Yevick approximation and the most recent simulation data.

Alternative Approaches to the Equilibrium Properties of Hard-Sphere Liquids

An overview of some analytical approaches to the computation of the structural and thermodynamic properties of single component and multicomponent hard-sphere fluids is provided. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions for the contact values of the radial distribution functions and the corresponding analytical equations of state are also discussed. Extensions of this methodology to related systems, such as sticky hard spheres and square-well fluids, as well as its use in connection with the perturbation theory of fluids are briefly addressed.