Numerical Solution of Advection-diffusion Equation Using a Sixth-order Compact Finite Difference Method

This study aims to produce numerical solutions of one-dimensional advection-diffusion equation using a sixth-order compact difference scheme in space and a fourth-order Runge-Kutta scheme in time. The suggested scheme here has been seen to be very accurate and a relatively flexible solution approach in solving the contaminant transport equation for Pe ≤ 5. For the solution of the present equation, the combined technique has been used instead of conventional solution techniques. The accuracy and validity of the numerical model are verified through the presented results and the literature. The computed results showed that the use of the current method in the simulation is very applicable for the solution of the advection-diffusion equation. The present technique is seen to be a very reliable alternative to existing techniques for these kinds of applications.


Introduction
Problems of environmental pollution can always be reduced to the solution of a mathematical model of advection diffusion. The unknown quantity in these cases is the concentration, , a scalar physical quantity, which represents the mass of a pollutant or the salinity or temperature of the water [1]. Advection-diffusion equation (ADE) illustrates many quantities such as mass, heat, energy, velocity, and vorticity [2]. The ADE has been used as a model equation in many engineering problems such as dispersion of tracers in porous media [3], pollutant transport in rivers and streams [4], the dispersion of dissolved material in estuaries and coastal seas [5,6], contaminant dispersion in shallow lakes [7], long-range transport of pollutants in the atmosphere [8], thermal pollution in river systems [9], and flow in porous media [10]. The advection-diffusion transport equation in one-dimensional case without source terms is as follows: with initial condition ( , = 0) = 0 ( ) (2) and boundary conditions ( = 0, ) = 0 ( ) , where is time, is space coordinate, is diffusion coefficient, ( , ) is concentration, ( , ) is velocity of water flow, and is length of the channel, respectively. Here 0 , 0 , , and are prescribed functions, whilst is the unknown function. Notice that > 0 and > 0 are considered to be positive constants quantifying the diffusion and advection processes, respectively.
Utility of higher-order numerical methods in solving many problems accurately is required. Lately, a noticeable interest in the development and application of CD methods for solving the Navier-Stokes [39][40][41] and other partial differential equations [42][43][44][45][46] has been renovated. Narrower stencils are required in the CD schemes, and by a comparison to classical difference schemes, they have less truncation error. In the current paper, accurate solutions of the ADE are obtained by using a sixth-order compact difference (CD6) [47], a fourth-order Runge-Kutta (RK4) schemes in space and time, respectively.

The Compact Finite Difference Method
CD methods are very popular in the fluid dynamics community because of their high accuracy and advantages associated with stencils [48]. These methods are efficient for higher accuracies without any increase in a stencil, while traditional high-order finite difference methods use larger stencil sizes that make boundary treatment hard. It can also be noted that the CD schemes have been demonstrated to be more precise and computationally economic. Use of smaller stencil sizes in the CD methods is useful when dealing with nonperiodic boundary conditions. A uniform one-dimensional mesh is considered, consisting of points: 1 , 2 , . . . , −1 , , +1 , . . . , with the mesh size ℎ = +1 − . The first-order derivatives of the unknown function can be given at interior nodes as follows [47]: leading to an -family of fourth-order tridiagonal schemes with A sixth-order tridiagonal scheme is obtained by = 1/3, Approximation formulae for the derivatives of nonperiodic problems can be derived with the consideration of onesided schemes for the boundary nodes. Interested readers are referred to the work of Lele [47] for details of the derivations for the first-and second-order derivatives.
The third-order formula at boundary point 1 is as follows: The fourth-order formula at boundary points 2 and − 1 is as follows: The third-order formula at boundary point is as follows: Use of the above formulae leads to following matrix equation: where = ( 1 , . . . , ) . Consideration of the first-order operator twice will give us the second-order derivative terms; that is, The RK4 scheme is considered to obtain the temporal integration in the present study. Utility of the CD6 technique to (1) gives rise to the following differential equation in time: where indicates a spatial linear differential operator. The spatial and temporal terms are approximated by the CD6 and the RK4 schemes, respectively. The semidiscrete equation (12) is solved using the RK4 scheme, through the following operations: To obtain the approximate solution of (1) with the boundary and initial conditions using the CD6-RK4, the domain [0, ] is first discretized such that 0 = 1 < 2 < ⋅ ⋅ ⋅ < = where is the number of grid points.

Numerical Illustrations
Let us consider the advection-diffusion equation with the initial and boundary conditions. The numerical results are compared with the exact solutions. The differences between the computed solutions and the exact solutions are shown in Tables 1-5. Three examples for which the exact solutions     are known are used to test the method described for solving the advection-diffusion equation. The technique is applied to solve the ADE with 0 ( ), 0 ( ), ( ), and ( ) prescribed.
To test the performance of the proposed method, 2 and ∞ error norms are used as follows: Example 1. Here, pure advection equation is considered in an infinitely long channel of constant cross-section and bottom slope, and velocity is taken to be = 0.5 m/s. Concentration is accepted to be the Gaussian distribution of = 264 m, and initial peak location is 0 = 2000 m. The initial distribution is transported downstream in a long channel without change in shape by the time = 9600 s. Exact solution of this problem is as follows [11]: At the boundaries the following conditions are used: Initial conditions can be taken from exact solution. The initial Gaussian pulse at = 0 , the concentration distribution obtained using the CD6 solution, and concentration  distribution obtained using exact solution at = 9600 s are compared in Figure 1. Both quantitative and qualitative agreements between the exact and the CD6 solutions are excellent (see Tables 1 and 2, Figure 1).
As seen from Table 1, the CD6 method has given the closest result to the exact peak concentration value. Note that, since an explicit time integration scheme RK4 is used in this study, the CD6 scheme cannot produce any result for Cr > 1.
To show the accuracy of the obtained results, 2 and ∞ error norms have been calculated using the CD6 scheme and exhibited in Table 2. As seen in the corresponding table, the CD6 solution is better than its rivals.
At the boundaries the following conditions are used: Initial conditions can be taken from exact solution. Comparison between numerical solutions and the exact solution is given in Table 3. In the calculation of the exact results given by Szymkiewicz [11], there has erroneously been a Mathematical Problems in Engineering mistake. Therefore the exact results have been recalculated in MATLAB. As seen in Table 3, the solutions produced by other researchers [11,28,31,32,34] for Cr = 0.6 do not converge enough. This case proves that the selected time step (Δ = 60 s) is greater than it needs to be. Note also that the CD6 scheme gives stable results for Cr = 0.6, and the computed results are nearly as good as in FEMQSF [34]. Since this problem cannot be solved accurate enough for Δ = 60 s, the calculations have been repeated for the cases Δ = 10 s (Cr = 0.1) and Δ = 1 s (Cr = 0.01), and the corresponding results have been given in Table 3. The results produced by the CD6 scheme for Δ = 1 s are the same as with the exact solution, while the results of the CD6 scheme for Δ = 10 s are seen to be acceptable level. Comparison of the exact solution and the numerical solution obtained with CD6 scheme for ℎ = 1 m and Δ = 10 s is shown in Figure 2. As can be seen in Figure 2, there is an excellent agreement between CD6 and exact solutions.  (1). The following exact solution for this example can be found in [49]: At the boundaries the following conditions are used:  distribution obtained using the CD6 solution as shown in Figure 3. As can be seen in Table 4, the CD6 results in Example 3 are far more accurate comparison to Crank-Nicolson (CN) scheme [12] and third-order finite difference (FD3) scheme [17]. In Table 5, a comparison of analytical and CD6 solutions is carried out for various values of with ℎ = 0.025, Δ = 0.005, Cr = 0.16, and Pe = 4.

Conclusions
This paper deals with the advection-diffusion equation using the CD6 scheme in space and the RK4 in time. The combined method worked very well to give very reliable and accurate solutions to these processes. The CD6 scheme provides an efficient and alternative way for modeling the advectiondiffusion processes. The performance of the method for the considered problems was tested by computing 2 and ∞ error norms. The method gives convergent approximations for the advection-diffusion problems for Pe ≤ 5. Note that numerical solution cannot obtained while Pe > 5 and Cr > 1. To overcome these disadvantages, upwind compact schemes and implicit time integration need to be used. For further research, special attention can be paid on the use of compact difference schemes in computational hydraulic problems such as sediment transport in stream and lakes, contaminant transport in groundwater, and flood routing in river and modeling of shallow water waves.