Numerical Solution of Advection Diffusion Equation Using Semi-Discretization Scheme

Numerical diffusion and oscillatory behavior characteristics are averted ap-plying numerical solutions of advection-diffusion equation are themselves im-mensely sophisticated. In this paper, two numerical methods have been used to solve the advection diffusion equation. We use an explicit finite difference scheme for the advection diffusion equation and semi-discretization on the spatial variable for advection-diffusion equation yields a system of ordinary differential equations solved by Euler’s method. Numerical assessment has been executed with specified initial and boundary conditions, for which the exact solution is known. We compare the solutions of the advection diffusion equation as well as error analysis for both schemes.


Introduction
The advection diffusion equation (ADE) is the model that can be used for simulation natural processes. Two categories of the advection-diffusion equation: advection is first due to the movement of materials from one region to another; the second category is called diffusion which is due to the movement of materials from higher concentration to low concentration. This mathematical model has a wide range of applications in natural science and engineering. These applications include where simulation techniques are useful for transport of air, river water, adsorption of pollutants in soil, food processing, modeling of the biological system, finance, electromagnetism, fluid mechanics structural dynamics, quantum physical process, etc. The analytical and numerical solutions along with an initial and two boundary conditions help to comprehend pollutant concentration distribution behavior through an open medium like rivers, air, lakes, and porous medium. Various works have been appeared to solve and use this equation in their simulation using finite difference methods [1] [2] [3]. Numerical Solution of the 1D advection-diffusion Equation solved using standard and nonstandard Finite Difference Schemes [4]. The significant application of the linear advection diffusion equation lies in fluid dynamics, heat transfer, and mass transfer [5]. Researchers examine numerical solution of Advection Diffusion Equation using operator splitting method [6]. These methods have been implemented by a characteristic method with cubic spline interpolation (MOC-CS) and Crank-Nicolson (CN) finite difference scheme. Obtained results were compared with analytical solutions. It is seen that the implemented method has lower error than other methods also produces accurate results even when the time steps are great. The linear advection diffusion equation (ADE) is a model which describes the contaminant transport due to the combined effect of advection and diffusion in a porous media [7]. In this study, the advection diffusion equation is solved by explicit finite difference schemes and investigates a different approach, the semi-discretization method: a spatial variable which yields a system of ODE with the temporal independent variable. We solve this system of ODE's by Euler's method and develop an algorithm of the Euler method for the system of ODE's and implement it for the computation of the concentration ( ) , u x t .

Advection Diffusion Equation
We consider the following partial differential equation, which has both an adventive and diffusive terms together.
with initial condition: ; And boundary conditions:

Numerical Methodology
In mathematics, our goal is to approximate the solution of the differential equations.  This gives a large algebraic system of equations to be solved in replace of the differential equation, which can be easily solved [11] [12] on a computer by Matlab code.

Computational Grid
We consider some simple space discretization on a uniform grid. We divide the are found by replacing the spatial derivatives by difference quotients. we also divide the time interval [ ] and T t N ∆ = . For purpose of the notation x h ∆ = and t k ∆ = .
This gives a finite difference discretization in space. Setting Therefore, we get a system of ordinary difference equations (ODEs) of (1.1) with a given initial value ( ) 0 u .

Discretization of Explicit Finite Difference Scheme (ECDS)
To approximate the solution to Equation (1) using the Explicit Centered Difference Scheme, we use the following approximations where x ∆ is the spatial step, t ∆ is the time step, m and n is spatial and temporal node respectively. Substituting Equation (a), (b), (c) in Equation (1)

Spatial Discretization Technique for ADE and Solved by Euler Method
We consider the following figure for ADE (Advection Diffusion Equation) .
We draw vertical grid line as shown in the picture. These lines are parallel to the t-axis and cross the x-axis in In semi-discretization method (SDM), we assume that the PDE system with its boundary conditions has spatially discretized, and thus we focus on ODE sys- , representing semi-discrete advection-diffusion problems.
We introduce the function of one variable: Now approximate the first derivative and second derivative x ∂ and xx ∂ respectively as: This gives the semi-discrete form of (1) So, we obtain a linear system of ordinary differential equations (ODE's) of the type.
, where and , are vector

Algorithm for the Semi Discretization of ADE Solved by Euler Method
To approximate the solution to the partial differential equation  Step 4-Step 6: (solve a tri-diagonal linear system) Step 4: Construct matrix A and B Set,

Case Study
Numerical implementation of Euler method [10] [11]: our solving equation (1): We can rewrite general nth term in Equation (4) to required Euler method of advection diffusion equation.  ; & 2 n n n n n n n n n n n P n n n

Semi Discretization Scheme for ADE Solved Using Euler Method
For this case the time step is increased to 0.008

Error Analysis
The comparison of relative error for two finite difference schemes as a function of time is shown in Figure 3. Two different curves show relative error for Explicit (red) and Euler (blue). We found the relative error for ECDS remains below 0.0009 and the relative error for Euler remains below 0.0011.       represents the concentration is high for 10 seconds and the green curve shows that concentration is decreased for 20 seconds. The curve marked by "blue"

Problem Discussion
represents concentration is high for 30 seconds and. The plot marked by "yellow" represents concentration is high for 50 seconds we can see that when time is increased concentration profile is decreasing. As can be seen in Figure 6, five    both velocity and diffusion coefficient at t = 20 secs, the solutions appeared which can be seen in Figure 8.
Similarly, concentration profile by solving numerically at a different diffusion rate can be seen in Figure 9 where the maximum concentration profile is shown at time 20 sec t = .

Conclusion
We present numerical solutions with exact solutions for the advection-diffusion equation with an initial condition and two boundary conditions by using ECDS and semi discretize method. A numerical study of the ADE has been presented graphically for two different schemes. The numerical solution of ADE by semi-discretization scheme shows a good agreement with the exact solution as well as for ECDS. Though, ECDS seems a more efficient scheme in terms of elapse timing; we compute the relative errors for two different schemes, both schemes show a very good rate of convergence. In comparison to Euler's method, ECDS shows less error, which is obvious. Semi discretize methods have to deal with a large number of systems of ordinary differential equations in comparison with ECDS. In our next work, we would like to upgrade this work with higher-order accuracy.