Numerical Simulation of the Fractional Dispersion Advection Equations Based on the Lattice Boltzmann Model

-e fractional dispersion advection equations (FDAEs) have recently attracted considerable attention due to their extensive application in the fields of science and engineering. For example, it has been shown that the anomalous solute transport behaviour that exists in hydrology can be well explained by introducing FDAEs.-erefore, the study of FDAEs has profound significance for understanding real transport phenomena in nature. Nevertheless, the existing algorithms for the FDAEs are generally intricate and costly. -erefore, exploiting an efficient solution technique has been a concern for scientists. In an effort to overcome this challenge, a promising lattice Boltzmann (LB) model for the FDAEs is presented in this paper. -e Riemann–Liouville definition and the Grünwald–Letnikov definition are introduced for the time derivatives. In addition, Chapman–Enskog analysis is applied to recover the FDAEs. To test the validity of the model, three numerical examples are carried out. In addition, a comparative study of the proposed model and the classical implicit finite difference scheme is also conducted. -e numerical results show that the model is suitable for simulating FDAEs.


Introduction
To overcome certain disadvantages of lattice gas automata, such as the lack of Galilean invariance and the undesired statistical noise, the LB method was developed by introducing statistically averaged particle distribution functions instead of Boolean variables. e method's popularity in recent years can be attributed to its numerical stability, algorithmic parallelism, and programming simplicity. As a mesoscopic numerical method, it has shown its strength in simulating complex flow systems [1][2][3].
e authors of the current work are more concerned about the LB approaches for FPDEs. To date, a number of related works have been documented in the literature. For example, Xia et al. developed a multispeed LB model for the FPDE and successfully simulated the anomalous superdiffusion phenomenon [34]. An LB model was presented for subdiffusion equations in 2016 [35]. e numerical examples proposed by Zhang and Yan showed that the scheme had first-order accuracy. Zhou et al. simulated fractional advection-diffusion equations based on the LB model [36]. e model with second-order accuracy successfully predicted the mass transport in hydrological systems. In 2018, Dai et al. constructed an LB model for space-fractional reaction-diffusion equations with nonlinear source terms [37]. Wang et al. proposed a two-time relaxation LB model to describe where c, ] ∈ (0, 1); A, B, and F are the parameters that need to be determined based on specific problems, and q(x, t) is the source term. From equation (1), we can clearly observe that both the dispersion term and the advection term involve time-fractional derivatives. is equation plays a significant role in anomalous diffusion problems [17,42]. However, the complexity of this FDAE makes most of the classical numerical algorithms inefficient and intricate. Exploiting an appropriate method for solving equation (1) is still an arduous task for scientists. e LB method has emerged as an active candidate in the past three decades and seems to be a promising tool to study this FDAE. However, there has been no relevant LB model that can handle this equation to date. Based on the above description, we intend to propose an effective LB algorithm to investigate the FDAE in this research.
To apply the LB model, the Riemann-Liouville definition is employed for the time-fractional derivative terms and is given by the following equation: (3b) erefore, equation (1) can be rewritten as follows: e integral form is sometimes inconvenient for solving numerically. Hence, we apply the Grünwald-Letnikov definition to convert equations (3a) and (3b) into a series form: e remaining part of the paper is constructed as follows. In Section 2, an LB model for FDAEs is presented. In Section 3, three numerical experiments are carried out to demonstrate the validity of the proposed model. In Section 4, based upon the results, conclusions are made and presented.

Model Formulation
For a one-dimensional LB equation with the Bhatnagar-Gross-Krook collision term, where f α (x, t) is the discrete distribution function, e α is the discrete velocity in the α direction, Δt represents the time step of the model, τ represents the relaxation time, and f eq α (x, t) is the local equilibrium distribution function, which should meet the conservation law given by Ω α (x, t) is called the additional term, which can be expanded as follows: where ε is a dimensionless expansion parameter that numerically equals Δt. erefore, equation (6) can be rewritten as It is feasible to assume that the time step of our model Δt � ε is a small parameter. erefore, we can employ the Taylor series expansion and Chapman-Enskog expansion [43]: where f (0) α � f eq α . In addition, we introduce t n as time scales: t n � ε n t, n � 0, 1, 2, . . . , Now, substituting equations (8), (10), (11), and (12b) into equation (9), separating and reorganizing the terms based on different orders of ε, we can derive the following equations: where the partial differential operator Δ � (z/zt 0 )+ e α (z/zx). C 2 is a polynomial of τ given as For the purpose of simulating FDAEs, the macroscopic variable u(x, t) should be defined as According to the definition of macroscopic variable equation (15), the conservation law given by equation (7), and the Chapman-Enskog expansion given by equation (11), we can derive e first two moments of f (0) α (x, t) are defined and selected as where the parameter η is given by en, the FDAE with second-order accuracy of the truncation error can be recovered: e detailed recovery process can be found in Appendix. Combining equations (16a), (17a), and (17b), the determination of equilibrium distribution functions is straightforward. In this paper, the D1Q3 LB model is employed, in which the discrete velocities are selected as where c represents the lattice speed. e solutions to equilibrium distribution functions are given by

Numerical Examples
Example 1. Consider the following one-dimensional FDAE: where 0 < c < 1 and A � 0.001. e variables x, t ∈ [0, 1]. e source term q(x, t) is denoted in the following equation: e initial and boundary conditions are given by e exact solution to the problem is given by By employing equation (5a), the initial and boundary conditions of variable u are given by

Mathematical Problems in Engineering
From equation (3a), the exact solution of variable u is obtained: In Figure 1(a), the exact solutions and LB numerical solutions for t � 1.0 are shown. e parameters are as follows: c � 0.5, total lattice number M � 100, Δx � 0.01, Δt � 0.01, c � 1, and τ � 0.908. e solutions of variables u(x, t) and u(x, t) are both presented, which illustrate that the LB solutions agree with the exact solutions.
In Figures 1(b) and 1(c), surface plots of the absolute and relative errors are presented, respectively. e definitions of the absolute errors and relative errors are given by equations (27a) and (27b), respectively: In equations (27a) and (27b), u N represents the solutions calculated using the LB model presented in this paper and u E represents the exact solutions.
To show the errors more clearly, we also plot the absolute and relative errors when t � 1.0 in Figure 1 To analyse the relationship between the errors and lattice size Δx, a log-log graph is plotted for t � 1.0 (see Figure 1(e)). e regression fit is also employed to quantify the trend of errors. e results show that log 10 E a � 1.008log 10 Δx + 0.339 and log 10 E r � 1.013log 10 Δx − 0.083, which indicate that the model's convergence order is 1.0 in space.
For comparison with the existing FDAE algorithm, we apply the following implicit difference scheme at the discrete point (x n , t k+1 ) [44]: e comparison of the infinity norm of absolute errors ‖E a ‖ ∞ for x � 0.5 between our model and the implicit difference scheme is shown in Table 1, where the infinity norm of absolute errors ‖E a ‖ ∞ is defined as follows: Here, in equation (29), T is the total discrete time; u E (x n , t k ) represents the exact solution at point (x n , t k ); u N (x n , t k ) represents the LB or difference result at discrete point (x n , t k ). Table 1 illustrates that the infinity norm of the absolute errors of the LB model proposed in this paper is larger than that of the implicit difference scheme in equation (28).
We also compute the convergence rate of the LB model and the implicit difference scheme for x � 0.5 and present the results in Table 2. Here, convergence rate R(Δx, Δt) is defined as e convergence rate can be used to measure the speed with which the numerical solution converges to the exact solution as the discrete points increase. From Table 2, we can observe that the convergence rates of the LB model and the implicit difference scheme are close to 2.0, which means that if the discrete points doubled, the infinity norm of the absolute errors will be nearly halved.
In addition, we introduce another coefficient ρ � ‖E a ‖ ∞ ((Δx, Δt)/(Δx)). Coefficient ρ represents the scaling factor of the infinity norm of the absolute errors ‖E a ‖ ∞ (Δx, Δt) and space step Δx. In Table 3, the parameter ρ for the LB model and the implicit difference scheme for x � 0.5 is given. From the table, it can be clearly seen that the coefficient ρ for the LB model is within the range of (1.27, 1.35), and for the implicit difference scheme, the coefficient ρ is within the range of (0.117, 0.124) in this example. With the adjustment of the time and space step, the coefficient ρ for the LB model and the implicit difference scheme does not obviously change, which indicates that both the LB models we presented and the implicit difference scheme have a first-order accuracy of the truncation error.
Finally, we list the computation time cost by these two numerical methods in Table 4. e LB model costs less CPU time because it is an explicit algorithm, which eliminates the trouble of solving linear algebraic equations. It can be foreseen that when the calculation scale increases, the characteristic of the LB model's low time cost will be more obvious. In particular, the LB model has the advantage of algorithmic parallelism, which is suitable for the large-scale calculation of FDAEs in practical engineering problems.
From the comparison performed above, we can see that the infinity norm of absolute errors ‖E a ‖ ∞ for the LB model is larger than that of the implicit difference scheme. e LB model significantly reduces the CPU time cost. In addition, these two numerical methods have the same convergence rate and convergence order. e LB model, as an explicit algorithm, has several advantages over the finite difference scheme (outstanding explicit algorithmic stability, high algorithmic parallelism, simple handling of complex boundaries, etc.). e LB model presented in this paper is an efficient and selectable method for the FDAE.
Example 2. In this example, the following FDAE is simulated: . (32) e initial condition is given by e Dirichlet boundary conditions are given by the following equations: e exact solution to this problem is given as follows: From equations (5a)-(5b), we can obtain the initial and boundary conditions of variables u(x, t) and u(x, t): By employing equation (3a) and (3b), we can derive the exact solutions of u(x, t) and u(x, t): e numerical results are shown in Figure 2. In Figure 2  ese two graphs show that the absolute errors are less than 3.0 × 10 − 2 , whereas the relative errors are consistently lower than 8.0 × 10 − 2 from t � 0.1 to t � 1.0. e relationship between the errors and spatial positions x for t � 1.0 is plotted in Figure 2(d). e parameters are the same as those shown in Figure 2(a). From Figure 2(d), we can observe that the absolute errors E a are less than 0.03, while the relative errors E r are close to 0.01 for t � 1.0.
Example 3. Finally, the following form of FDAE is considered: e parameters are A � 0.001 and 0 < c < 1. e variables x, t ∈ [0, 1]. e source term is q(x, t) (see equation (39) for their relationship): e initial condition of the problem is given by e Dirichlet boundary conditions are given by e exact solution to the problem is By applying equations (5a) and (3a) as mentioned in Example 1 and Example 2, we can obtain the initial and boundary conditions of u(x, t) as well as the exact solutions of u(x, t): In Figure 3 When t � 1.0, E a < 0.001 and E r < 0.02. e numerical results presented above imply that our model is suitable for simulating FDAEs.

Conclusions
In this paper, an LB model is proposed to investigate the FDAE. e Riemann-Liouville definition and the Grünwald-Letnikov definition are applied to the timefractional derivatives. en, a simple LB model is presented to solve the equation. By introducing the Chapman-Enskog analysis, the FDAE with second-order accuracy of the truncation error is recovered. e model is efficient and promising for future practical applications, not only because the introduction of the Grünwald-Letnikov definition simplifies the calculations of time-fractional derivatives but also because the model retains advantages of the LB method over traditional numerical methods. ree numerical examples are carried out to test the validity of our model. e close agreement between the numerical results and the exact solutions implies that our model is suitable for simulating FDAE. In addition, a comparative study is also conducted. e results demonstrate that the proposed model and the implicit finite difference scheme have the same convergence rate and convergence order.
e LB model has a larger absolute error but costs less CPU time than the implicit finite difference scheme.
However, some aspects of the model are less successful and still need further research. e accuracy of the model is unsatisfactory. We think there are two possible reasons for the low spatial accuracy: (1) the treatment of the fractional derivative terms and (2) the introduction of additional distribution functions Ω (n) α . We are still looking for a better way to improve the accuracy of the Grünwald-Letnikov definition. In addition, constructing more effective and accurate additional distribution functions for the source terms in the FDAEs is an ongoing challenge and still needs further research.