Dimensional Diffusion Equation Research Articles

Multiple-Relaxation-Time Lattice Boltzmann scheme for fractional advection–diffusion equation

Partial differential equations (p.d.e) equipped with spatial derivatives of fractional order capture anomalous transport behaviors observed in diverse fields of Science. A number of numerical methods approximate their solutions in dimension one. Focusing our effort on such p.d.e. in higher dimension with Dirichlet boundary conditions, we present an approximation based on Lattice Boltzmann Method with Bhatnagar–Gross–Krook(BGK) or Multiple-Relaxation-Time (MRT) collision operators. First, an equilibrium distribution function is defined for simulating space-fractional diffusion equations in dimensions 2 and 3. Then, we check the accuracy of the solutions by comparing with (i) random walks derived from stable Lévy motion, and (ii) exact solutions. Because of its additional freedom degrees, the MRT collision operator provides accurate approximations to space-fractional advection–diffusion equations, even in the cases which the BGK fails to represent because of anisotropic diffusion tensor or of flow rate destabilizing the BGK LBM scheme.

Analysis of the Numerical Solution in the One Dimensional Non-Stationary Diffusion Equation using Schemes of Finite Mimetic Differences and Crank Nicolson

espanolSe propone la solucion numerica de la ecuacion de difusion no estatica unidimensional, desarrollando un algoritmo en software Matlab version 7.0, para lo cual se combina el esquema de diferencias finitas mimeticas en la aproximacion de los operadores diferenciales del continuo (gradiente y divergencia) para la variable espacial, sobre una malla uniforme, cuyos operadores diferenciales discretos presentan una aproximacion de segundo orden y el enfoque en diferencias finitas tipo Crank-Nicolson para obtener aproximaciones en la variable temporal. Este algoritmo propuesto para los enfoques mimeticos y Crank-Nicolson presentan mejor aproximacion que el esquema en diferencias finitas tipo Crank- Nicolson. Ademas se calcula el error de aproximacion generado entre la solucion numerica y la solucion analitica usando la norma del maximo para la ecuacion de difusion no estacionaria con condiciones de frontera tipo Robin. EnglishThe numerical solution of the one-dimensional non-static diffusion equation is proposed, developing an algorithm in software Matlab version 7.0, for which the mimetic finite difference scheme is combined in the approximation of the differential operators of the continuum (gradient and divergence) for the spatialvariable, on a uniform grid, whose discrete differential operators have a second order approximation and the finite difference approach type Crank Nicolson for approximations in the temporary variable.This proposed algorithm for the mimetic and Crank Nicolson approaches has a better approximation than the finite-difference Crank Nicolson-type scheme.In addition, the approximation error generated between the numerical solution and the analytical solution is calculated using the maximum standard for the non-stationary diffusion equation with Robin type boundary conditions.

English

This paper examines a general recurrence relation by the use of fractional reduced differential transform and then a scheme (methodology) on how to find closed solutions of one dimensional time fractional diffusion equations with initial conditions in the form of infinite fractional power series and in terms of Mittag-Leffler function in one parameter as well as their exact solutions by the use of fractional reduced differential transform method. The new general recurrence relation and the methodology of the fractional reduced differential transform method were successfully developed. The obtained new general recurrence relation helps us to solve time-fractional diffusion equations with initial conditions and various external forces by using fractional reduced differential transform method. To see its effectiveness and applicability, five test examples were presented. The results show that the general recurrence relation works successfully in solving time-fractional diffusion equations in a direct way without using linearization, transformations, perturbation, discretization or restrictive assumptions by using fractional reduced differential transform method.   Key words: Time fractional diffusion equations with initial conditions, Caputo fractional derivatives, Mittag-Leffler function, Fractional reduced differential transform method (FRDTM).

Timescale Monitoring of Vesuvian Eruptions Using Numerical Modeling of the Diffusion Equation

It is necessary to understand the diffusive behavior of the volatile phase in order to interpret the control mechanisms of the bubble size in volcanic rocks and the processes influencing the eruption itself. A simple numerical model is proposed, based on the time-dependency of the diffusion equation for a hollow sphere, to simulate incorporation of atmospheric noble gases in pumice for Vesuvian (i.e., Plinian) eruptions. During a Vesuvian eruption, melt fragments are ejected into the air. The resulting pumice samples collected after the eruption phase exhibit a significant incorporation of elementally and isotopically fractionated atmospheric noble gases. The noble gas content of the trapped gases then potentially provides useful data with which to constrain the timescales of cooling for the samples. The system can be adequately described as a diffusion process into a hollow sphere through the one dimensional diffusion equation for a spherically symmetrical geometry. Diffusion coefficients are time-dependent to include the effect of an exponential decay of temperature with time. The complexity of this system requires numerical resolution of the diffusion equation due to the diffusion coefficient temperature dependency. The outer boundary condition is fixed with a given noble gas concentration via an inhomogeneous Dirichlet boundary condition, while the inner boundary condition is set as a flux-free boundary. The numerical model allows the noble gas content, and thus the noble gas elemental and isotopic ratios entering the bubble, to be modeled as a function of time, hollow sphere thickness, diffusion coefficients, initial and final temperatures, and quench rate.

On the equivalence groups for (2+1) dimensional nonlinear diffusion equation

(2+1) dimensional diffusion equation is considered within the framework of the group of equivalence transformations. Generators for the group are obtained and admissible transformations between linear and nonlinear equations are examined. It is shown that transformations between linear and nonlinear equations are possible provided that the generators of independent variables depend on the dependent variable. Exact solutions for some nonlinear equations are obtained.

Finite Difference Formulation for Prediction of Water Pollution

Water is an important component of the earth. Human being and living organisms are demand for the quality of water. Human activity is one of the causes of the water pollution. The pollution happened give bad effect to the physical and characteristic of water contents. It is not practical to monitor all aspects of water flow and transport distribution. So, in order to help people to access to the polluted area, a prediction of water pollution concentration must be modelled. This study proposed a one-dimensional advection diffusion equation for predicting the water pollution concentration transport. The numerical modelling will be produced in order to predict the transportation of water pollution concentration. In order to approximate the advection diffusion equation, the implicit Crank Nicolson is used. For the purpose of the simulation, the boundary condition and initial condition, the spatial steps and time steps as well as the approximations of the advection diffusion equation have been encoded. The results of one dimensional advection diffusion equation have successfully been used to predict the transportation of water pollution concentration by manipulating the velocity and diffusion parameters.

The fractional Dodson diffusion equation: a new approach

In this paper, after a brief review of the general theory concerning regularized derivatives and integrals of a function with respect to another function, we provide a peculiar fractional generalization of the $(1+1)$-dimensional Dodson's diffusion equation. For the latter we then compute the fundamental solution, which turns out to be expressed in terms of an M-Wright function of two variables. Then, we conclude the paper providing a few interesting results for some nonlinear fractional Dodson-like equations.

A high accuracy scheme of three layers and nine points for Solving the concentration diffusion equation

A three layers and nine points implicit difference scheme is proposed for solving the one dimensional diffusion equation, using the method of undetermined coefficients and the accuracy is . Though the numerical solution and the Von Neumann analysis, the stability condition is proved to be . The stability condition and accuracy of the scheme are verified by numerical calculation. Compared with the existing three—layers scheme, this paper’s scheme reaches a higher accuracy in case of less nodes. In summary, the scheme mentioned in this paper has academic significance and application value.

A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems

In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa’s HOC ADI method [26], Cao and Ge’s HOC ADI method [27], and our previous exponential HOC ADI method [28].

Fractional Diffusion Models for the Atmosphere of Mars

The dust aerosols floating in the atmosphere of Mars cause an attenuation of the solar radiation traversing the atmosphere that cannot be modeled through the use of classical diffusion processes. However, the definition of a type of fractional diffusion equation offers a more accurate model for this dynamic and the second order moment of this equation allows one to establish a connection between the fractional equation and the Ångstrom law that models the attenuation of the solar radiation. In this work we consider both one and three dimensional wavelength-fractional diffusion equations, and we obtain the analytical solutions and numerical methods using two different approaches of the fractional derivative.

Tungsten-deuterium co-deposition: Experiment and analytical description

Deuterium to tungsten ratio in co-deposited films was measured in a wide range of deposition temperatures from room temperature to 800 K. The experimental curve had 3-4 steps of decrease, which cannot be described by empirical scaling known from literature. A simple analytical model based on 1-dimensional diffusion equation with trapping was developed. A good agreement of the model and the experiment was obtained suggesting three types of traps in W-D co-deposited layers with detrapping energies of 0.74 ± 0.08 eV, 1.18 ± 0.12 eV, and 1.56 ± 0.16 eV and atomic concentration of traps of 2.3%, 1.8%, and 1.0%, respectively.

Dispersion in two dimensional channels—the Fick–Jacobs approximation revisited

We examine the dispersion of Brownian particles in a symmetric two dimensional channel, this classical problem has been widely studied in the literature using the so called Fick–Jacobs’ approximation and its various improvements. Most studies rely on the reduction to an effective one dimensional diffusion equation, here we derive an explicit formula for the diffusion constant which avoids this reduction. Using this formula the effective diffusion constant can be evaluated numerically without resorting to Brownian simulations. In addition, a perturbation theory can be developed in where h0 is the characteristic channel height and L the period. This perturbation theory confirms the results of Kalinay and Percus (2006 Phys. Rev. E 74 041203), based on the reduction, to one dimensional diffusion are exact at least to . Furthermore, we show how the Kalinay and Percus pseudo-linear approximation can be straightforwardly recovered. The approach proposed here can also be exploited to yield exact results in the limit , we show that here the diffusion constant remains finite and show how the result can be obtained with a simple physical argument. Moreover, we show that the correction to the effective diffusion constant is of order and remarkably has some universal characteristics. Numerically we compare the analytic results obtained with exact numerical calculations for a number of interesting channel geometries.

Monte-Carlo simulation of a stochastic differential equation

For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient, Monte Carlo (MC) techniques are considered to be more effective than other algorithms, such as finite element method or finite difference method. The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques. For better convergence, methods with higher orders have been kept forward to allow MC codes with large step size. The main focus of this work is to look for operators that can produce converging results for large step sizes. As a first step, our comparative analysis has been applied to a general stochastic problem. Subsequently, our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.

An extension of the singular boundary method for solving two dimensional time fractional diffusion equations

In this paper, singular boundary method (SBM) in conjunction with the dual reciprocity method (DRM) is extended to the solution of constant and variable order time fractional diffusion equations (TFDEs). In this procedure, finite difference method breaks down the time domain and reduces the time fractional diffusion equation into a sequence of boundary value problems in inhomogeneous Helmholtz-type equations. Then SBM-DRM is applied to space semi-discretization of these types of equations, in a two step process. First, DRM, which is a popular meshless method based on radial basis functions (RBFs), is applied to obtain the particular solution. After evaluating the particular solution, singular boundary method can be employed to evaluate the homogeneous solution. To consider the accuracy and efficiency of the presented method, some benchmark problems subjected to the Dirichlet and Neumann boundaries are examined on regular and irregular geometries.

Computational solutions of three-dimensional advection-diffusion equation using fourth order time efficient alternating direction implicit scheme

To develop an efficient numerical scheme for three-dimensional advection diffusion equation, higher order ADI method was proposed. 2nd and fourth order ADI schemes were used to handle such problem. Von Neumann stability analysis shows that Alternating Direction Implicit scheme is unconditionally stable. The accuracy and efficiency of such schemes was depicted by two test problems. Numerical results for two test problems were carried out to establish the performance of the given method and to compare it with the others Typical methods. Fourth order ADI method were found to be very efficient and stable for solving three dimensional Advection Diffusion Equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

Limit theorems for Lévy walks in d dimensions: rare and bulk fluctuations

We consider super-diffusive Lévy walks in dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of the central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have a universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called ‘infinite density’, describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of the PDF of velocity in one step. A typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Lévy distribution) interspersed with long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distributions the d-dimensional process can be reduced to a one-dimensional Lévy walk. We briefly discuss the consequences of non-universality for the d > 1 dimensional fractional diffusion equation, in particular the non-uniqueness of the fractional Laplacian.

Computational Solutions of Two Dimensional Convection Diffusion Equation Using Crank-Nicolson and Time Efficient ADI

To develop an efficient numerical scheme for two-dimensional convection diffusion equation using Crank-Nicholson and ADI, time-dependent nonlinear system is discussed. These schemes are of second order accurate in apace and time solved at each time level. The procedure was combined with Iterative methods to solve non-linear systems. Efficiency and accuracy are studied in term of L2, L∞ norms confirmed by numerical results by choosing two test examples. Numerical results show that proposed alternating direction implicit scheme was very efficient and reliable for solving two dimensional nonlinear convection diffusion equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

Numerical Study of Fisher’s Equation by Finite Difference Schemes

This research paper represents a numerical approximation to three interesting equations of Fisher, which are linear, non-linear and coupled linear one dimensional reaction diffusion equations from population genetics. We studied accuracy in term of L∞ error norm by random selected grids along time levels for comparison with exact results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension non-linear reaction diffusion equations.

Fast ADI method for high dimensional fractional diffusion equations in conservative form with preconditioned strategy

In this paper, high dimensional two-sided space fractional diffusion equations, derived from the fractional Fick’s law, and with monotonic variable diffusion coefficients, are solved by alternating direction implicit method. Each linear system corresponding to each spatial direction thus resulted is solved by Krylov subspace method. The method is accelerated by applying an approximate inverse preconditioner, where under certain conditions we showed that the normalized preconditioned matrix is equal to a sum of identity matrix, a matrix with small norm, and a matrix with low rank, such that the preconditioned Krylov subspace method converges superlinearly. We also briefly present some fast algorithms whose computational cost for solving the linear systems is O(nlogn), where n is the matrix size. The results are illustrated by some numerical examples.

Monotone Finite Volume Scheme for Three Dimensional Diffusion Equation on Tetrahedral Meshes

AbstractWe construct a nonlinear monotone finite volume scheme for three-dimensional diffusion equation on tetrahedral meshes. Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme, we present a new efficient eliminating method. The scheme has only cell-centered unknowns and can deal with discontinuous or tensor diffusion coefficient problems on distorted meshes rigorously. The numerical results illustrate that the resulting scheme can preserve positivity on distorted tetrahedral meshes, and also show that our scheme appears to be approximate second-order accuracy for solution.