907 Words4 Pages

Abstract:

The paper presents a formal exact solution of the one dimensional advection-diffusion equation (ADE) with constant coefficients in a finite domain. The numerical solution of the one-dimensional advection diffusion equation is obtained by using explicit centered difference schemes with FTBSCS and FTCS techniques for prescribed initial and boundary data. Numerical results for both the schemes are compared in terms of accuracy by error estimation with respect to exact solution of the and also, the numerical features of the rate of convergence are presented graphically.

Keywords: Advection Diffusion Equation, Exact solution, Explicit Finite Difference Schemes, Rate of Convergence.

1. Introduction

Advection–diffusion equation (ADE)*…show more content…*

Due to the growing surface and subsurface hydro-environment degradation and the air pollution, the advection–diffusion equation has drawn significant attention of hydrologists, civil engineers and mathematical modelers. The analytical/numerical solutions along with an initial condition and two boundary conditions help to understand the contaminant or pollutant concentration distribution behavior through an open medium like air, rivers, lakes and porous medium like aquifer. It has wide applications in other disciplines too, like soil physics, petroleum engineering, chemical engineering and*…show more content…*

We describe Mathematical Models for comparison of the error based on ([4], [5], [6], [7], [8], [9], [14], [15]). Based on the study of the general finite difference method for the second order linear partial differential equation ( [1], [2], [3], [11], [12], [13],[16], [17]), we develop an explicit finite difference scheme for our model treated as ADE as an IBVP with two sided boundary conditions in section 3. In section 3, we also establish the stability condition of the numerical schemes. In section 4, we present an algorithm for the numerical solution and we develop a computer programming code for the implementation of the numerical schemes and perform numerical simulations in order to verify the behavior for various parameters. In section 5, a comparison of errors for both the techniques with respect to an exact solution is projected herein in terms of accuracy. Numerical features of the rate of convergence are also presented graphically. Finally the conclusions of the paper are given in the last section.

2. Advection Diffusion Equation

The one-dimensional advection-diffusion equation [1] is given as

∂C/∂t+u ∂C/∂x=D (∂^2 C)/(∂x^2 )………(1) where C represents the solute concentration [ML-3] at x along longitudinal direction at time t, D is the solute dispersion, if it is independent of position and time, is called dispersion coefficient [L2T-1], t = time [T]; x = distance [L] and,

The paper presents a formal exact solution of the one dimensional advection-diffusion equation (ADE) with constant coefficients in a finite domain. The numerical solution of the one-dimensional advection diffusion equation is obtained by using explicit centered difference schemes with FTBSCS and FTCS techniques for prescribed initial and boundary data. Numerical results for both the schemes are compared in terms of accuracy by error estimation with respect to exact solution of the and also, the numerical features of the rate of convergence are presented graphically.

Keywords: Advection Diffusion Equation, Exact solution, Explicit Finite Difference Schemes, Rate of Convergence.

1. Introduction

Advection–diffusion equation (ADE)

Due to the growing surface and subsurface hydro-environment degradation and the air pollution, the advection–diffusion equation has drawn significant attention of hydrologists, civil engineers and mathematical modelers. The analytical/numerical solutions along with an initial condition and two boundary conditions help to understand the contaminant or pollutant concentration distribution behavior through an open medium like air, rivers, lakes and porous medium like aquifer. It has wide applications in other disciplines too, like soil physics, petroleum engineering, chemical engineering and

We describe Mathematical Models for comparison of the error based on ([4], [5], [6], [7], [8], [9], [14], [15]). Based on the study of the general finite difference method for the second order linear partial differential equation ( [1], [2], [3], [11], [12], [13],[16], [17]), we develop an explicit finite difference scheme for our model treated as ADE as an IBVP with two sided boundary conditions in section 3. In section 3, we also establish the stability condition of the numerical schemes. In section 4, we present an algorithm for the numerical solution and we develop a computer programming code for the implementation of the numerical schemes and perform numerical simulations in order to verify the behavior for various parameters. In section 5, a comparison of errors for both the techniques with respect to an exact solution is projected herein in terms of accuracy. Numerical features of the rate of convergence are also presented graphically. Finally the conclusions of the paper are given in the last section.

2. Advection Diffusion Equation

The one-dimensional advection-diffusion equation [1] is given as

∂C/∂t+u ∂C/∂x=D (∂^2 C)/(∂x^2 )………(1) where C represents the solute concentration [ML-3] at x along longitudinal direction at time t, D is the solute dispersion, if it is independent of position and time, is called dispersion coefficient [L2T-1], t = time [T]; x = distance [L] and,

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