1. Introduction
The transport of solutes by water takes place in a large variety of environmental, agricultural, and industrial conditions. An accurate prediction of the pollutant transport is crucial to the effective management of these systems [
1]. In the deterministic approach with constant transport parameters with respect to time and position the advection-diffusion equation (ADE) is linear and explicit closed-form solutions can generally be derived. Because of the large variability of flow and transport properties in the field, the often transient nature of the flow regime, and the non-ideal nature of applicable initial and boundary conditions, the usefulness of analytical solutions is often limited and numerical methods may be needed [
2].
A variety of numerical methods have been proposed for solving ADE, such as the method of characteristics [
3,
4,
5,
6,
7,
8,
9], the finite difference method [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19], the finite element method [
20,
21,
22,
23], the differential quadrature method [
24,
25], the Lattice Boltzman method [
26], and the meshless method [
27,
28,
29,
30]. In these studies, solutions of the ADE with constant parameters have been obtained. However, field and laboratory scale experiments indicate that the dispersivity in subsurface transport problems may be space- and time-dependent [
31,
32,
33,
34,
35].
Several numerical solutions of the ADE with variable parameters have been proposed in the literature. A numerical solution for one-dimensional transport of conservative and non-conservative pollutants in an open channel with steady unpolluted lateral inflow uniformly distributed over the whole length of the channel was presented by Ahmad [
36]. In Ahmad [
36], the flow velocity was considered to be proportional to the distance and the diffusion parameter proportional to the square of velocity to account for the lateral inflow into the channel. The governing equation was converted into an equation with constant parameters using a transformation approach. The transformed equation was then solved using a cubic spline interpolation and the Crank-Nicolson finite difference scheme for advection and diffusion parts, respectively. The computed results were compared with the analytical solution for a continuous injection of the pollutant at the upstream boundary. Ahmed [
37] developed a finite difference scheme based on a mathematical combination of the Siemieniuch and Gradwell approximation of time and the Dehghan’s approximation of the space variable. The author stated that the results were compared with analytical solutions and showed a good agreement. Savović and Djordjevich [
38] proposed an explicit finite difference method for the numerical solution of the one-dimensional ADE with variable parameters in semi-infinite media. The continuous point source of uniform nature was considered at the origin of the medium. Savović and Djordjevich [
39] then solved the same equation with a uniform pulse type input condition and the initial solute concentration that decreased with distance. In both studies, numerical solutions were obtained using the first-order explicit time integration approach and results were compared with analytical solutions reported in the literature. Gharehbaghi [
40] used the differential quadrature method to obtain the numerical solution of the ADE with variable parameters in the semi-infinite domain. Numerical solutions were obtained using the first-order explicit and implicit time integration approaches. To examine the accuracy and the efficiency of the suggested explicit and implicit differential quadrature approaches, obtained solutions were compared with solutions of the finite difference method. The author stated that numerical predictions of the implicit form gave better results than the explicit one. Gharehbaghi [
41] used the third- and fifth-order finite volume schemes [
42] to solve a time-dependent, one-dimensional ADE with variable parameters in a semi-infinite domain. Numerical solutions were obtained using the first-order explicit time integration approach. The results of the two schemes were compared with the performance of the QUICK scheme [
43]. The author stated that numerical predictions of the fifth-order finite volume scheme gave better results than the third-order finite volume and the QUICK schemes.
As can be seen from the afore-mentioned studies, low-order time integration schemes have been used in solving the ADE with variable parameters. To improve the accuracy of numerical solutions, it is required to use higher-order schemes in both time and space. The method of lines (MOL) is a well-established numerical procedure where the spatial derivatives in the partial differential equation (PDE) are approximated algebraically. Thanks to the MOL approach, a PDE system is converted into an ordinary differential equation (ODE) system. This ODE system is then integrated using standard numerical integration routines. Important variations of the MOL are possible. For example, the PDE spatial derivatives can be approximated using finite difference, finite element, finite volume, weighted residual method, and spectral method [
44]. In this study, we propose another variation of the MOL approach for solving ADE with constant and variable parameters. Spatial derivatives in the proposed MOL approach are approximated using a sixth-order combined compact difference scheme. The explicit fourth order Runge–Kutta method (ERK4) is used as the numerical integration routine in the MOL approach.
2. Advection-Diffusion Equation
Solute transport through a medium is described using a PDE of the parabolic/hyperbolic type. It is derived on the principle of conservation of mass and Fick’s laws of diffusion. This equation is usually known as the ADE. The one-dimensional ADE may be written in subscript notation as [
45]:
where
C is the solute concentration, D is the diffusion parameter and
U is the flow velocity at a position
x along the longitudinal direction at time
t. In Equation (1), D and
U can be rewritten as:
In the above equation,
and
may be referred to as the initial diffusion parameter and the uniform velocity, respectively.
and
are constants whose dimensions depend upon the expressions
and
. The initially solute-free state of the semi-infinite medium implies the following initial condition:
Because a continuous input concentration is introduced at the origin, whereas the concentration gradient at infinity is assumed to be zero, the following boundary conditions are obtained:
4. Numerical Applications
In this section, numerical solution of the ADE with constant and variable parameters are tested on three numerical examples. In these examples, input parameters are , and . The transport domain is divided into intervals of a constant length . Calculated concentration values are presented in longitudinal region at different times. Since the Neumann boundary condition is valid at the right boundary point, a large enough domain length is selected in all examples, i.e., . The numerical examples presented in this study are closely related each other. The initial and boundary conditions, the domain lengths , and the input parameters are the same. The parameters and , which control only temporal and spatial change, and the total simulation times are different. The main goal of this approach is to observe how the behavior of the problem changes when the velocity and diffusion coefficients are selected as variables. If the velocity and diffusion coefficients are fixed, the second and the third problem is reduced to the first problem.
4.1. ADE with Constant Parameters
The advection–diffusion equation is solved for a transport domain in which the velocity and the diffusion coefficient are constant, i.e.,
and
. These conditions describe the propagation of a steep front, which is simultaneously subjected to the diffusion. The analytical solution of the ADE with constant parameters is given by Szymkiewicz [
3]:
In this example, a numerical solution of an ADE with constant parameters and sharp behavior was performed. The analytical solution of this example has been calculated incorrectly by various researchers [
3,
20,
22]. Therefore, the results of these studies were not used in the comparison. Recently, Irk et al. [
58] was conducted a study, by approximating the spatial derivatives with cubic B-spline collocation scheme and extended cubic B-spline collocation. They were adopted the Crank–Nicolson scheme for the time integration. These two schemes will be referred to as CN-CBSC and CN-ECBSC.
norm errors of the ERK4-NCCD, CN-CBSC, and CN-ECBSC schemes for different time steps at
t are compared in
Table 1. In this example, the Peclet number
was selected as 5. The Courant numbers
were calculated as 0.01, 0.05, 0.1, 0.2, 0.3, 0.6, and 1, respectively, when the time steps used were ordered from the smallest to the largest. As can be seen from
Table 1, the ERK4-NCCD scheme is more accurate and more stable than the CN-CBSC and the CN-ECBSC schemes.
Table 2 shows the numerical results obtained with ERK4-NCCD at
t = 3000 s using different time steps. It can be observed from
Table 2, the ERK4-NCCD scheme it produces the more accurate solutions for
,
and
and gives acceptable solutions for larger time steps.
Table 3 shows the numerical solutions of ERK4-NCCD at
s,
s, and
s using the time step
. As seen from the
Table 3, there is almost no difference between the analytical and numerical solutions.
Numerical and analytical solutions of the ADE with constant parameters for
s is presented in
Figure 1.
Figure 1 shows that the numerical results obtained for
s coincide very well with the analytical solution.
Table 2 shows the numerical results obtained with ERK4-NCCD at
t = 3000 s using different time steps. It can be observed from
Table 2, the ERK4-NCCD scheme it produces the more accurate solutions for
,
and
and gives acceptable solutions for larger time steps.
4.2. ADE with Spatially Variable Parameters
In this example, we assumed that the diffusion coefficient is proportional to the square of the velocity, i.e.,
and
. Under this assumptions, the analytical solution of the ADE with spatially variable parameters is given by Kumar et al. [
59] as follows:
where
and
.
In this example, the velocity and the diffusion coefficient increase depending on the value of
. The numerical results at
s for different time steps and
are presented in
Table 4.
Table 4 shows that the ERK4-NCCD scheme was able to produce acceptable solutions for the maximum time step
s. When the simulation parameters were increased, there was a slight decrease in the stability of the scheme according to the previous example. Certainly, this inference is also valid for other numerical methods.
Figure 2 depicts the analytical and numerical solution of the ADE with spatially dependent parameters at
s for
s and
. These results can be compared with the problem presented above for
and
s in
Figure 1. When we compare
Figure 1 and
Figure 2, we observe that a change in
leads to a significant change in the solution of the problem. As can be observed in
Table 5, this change becomes more evident when we significantly increase
. The
norm errors of the ERK4-NCCD scheme for solving ADE with different
values at
s are provided in
Table 5.
4.3. ADE with Spatially and Temporally Variable Parameters
In this example, we assumed that the diffusion is proportional to square of the velocity parameter as considered second example. In addition, the diffusion and velocity are supposed to vary with temporally in the same proportion as considering
and
. Under this assumptions, the analytical solution of the ADE with spatially and temporally variable parameters is given by Kumar et al. [
59] as following:
where
,
,
,
and
is a dummy variable. In this example, the velocity and diffusion parameters are slightly increased depending on the
and
.
norm errors of the ERK4-NCCD scheme for
and different
values at
can be found in
Table 6. Increasing the
value leads to increasing Courant number,
. Therefore, we can face with stability problems for explicit time integration methods like ERK4. Referring to
Table 6, it is seen that the stability range decreased for
and
. The numerical results of the ERK4-NCCD scheme with
and
at
given in
Table 7. As can be seen from the table, quite less
and
values were obtained for all
values. Analytical and numerical solution of the ADE with spatially and temporally variable parameters at
for
, and
are presented in
Figure 3.
5. Conclusions
In this study, a combined compact finite difference scheme based on the method of lines is proposed for the numerical solution of the solute transport equation with variable parameters. In this approach, the partial differential equation representing the ADE is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Numerical examples with sharp concentration fronts are presented to demonstrate the effectiveness of the method. In the first example, we compared the numerical results of the ERK4-NCCD scheme with CN-CBSC and CN-ECBSC. It is found that the ERK4-NCCD scheme is more accurate and more stable than the CN-CBSC and the CN-ECBSC schemes. It was observed that the numerical results obtained using small time steps and the analytical results coincided with each other. Moreover, acceptable results were obtained when large time steps were used as well. The ERK4-NCCD scheme gives excellent results for long time-integration, i.e., t = 2000 s, 4000 s, and 6000 s. In second example, the velocity and diffusion coefficients were slightly increased depending on the value of . When the parameters increased, there was a slight decrease in the stability of the scheme compared to the previous example. We observe that a small change in leads to a significant change in the solution of the problem. This change becomes more evident when we significantly increase the . In the third example, we observed that increasing value leads to increasing Courant number. Therefore, we can face with stability problems for explicit time integration methods like ERK4. As a result, the proposed method has produced more accurate and more stable results than CN-CBSC and the CN-ECBSC schemes in solving ADE with constant and variable parameters. The proposed scheme produces fairly accurate results for the and the . It is recommended to use compact upwind schemes for higher Peclet numbers. When the method given in Gurarslan (2014) is followed, the proposed scheme can be easily extended to the two- and three-dimensional ADE.