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The 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. Therefore, 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. Therefore, 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. The 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. The numerical results show that the model is suitable for simulating FDAEs.

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. The 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 [

In addition, for nonlinear partial differential equations, the LB method has also become an effective numerical solver. LB models have widely been studied for various equations, including wave equations [

Fractional partial differential equations (FPDEs) have gained extensive attention in the field of scientific research due to their applications in physical systems [

The 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 [

In this paper, the following one-dimensional FDAE of the form given by equation (

To apply the LB model, the Riemann–Liouville definition is employed for the time-fractional derivative terms and is given by the following equation:

Denote

Therefore, equation (

The integral form is sometimes inconvenient for solving numerically. Hence, we apply the Grünwald–Letnikov definition to convert equations (

The remaining part of the paper is constructed as follows. In Section

For a one-dimensional LB equation with the Bhatnagar–Gross–Krook collision term,

It is feasible to assume that the time step of our model

Now, substituting equations (

For the purpose of simulating FDAEs, the macroscopic variable

According to the definition of macroscopic variable equation (

The first two moments of

Then, the FDAE with second-order accuracy of the truncation error can be recovered:

The detailed recovery process can be found in Appendix.

Combining equations (

Consider the following one-dimensional FDAE:

The initial and boundary conditions are given by

The exact solution to the problem is given by

By employing equation (

From equation (

In Figure

In Figures

In equations (

To show the errors more clearly, we also plot the absolute and relative errors when

To analyse the relationship between the errors and lattice size

For comparison with the existing FDAE algorithm, we apply the following implicit difference scheme at the discrete point

The comparison of the infinity norm of absolute errors

Here, in equation (

We also compute the convergence rate of the LB model and the implicit difference scheme for

The 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

In addition, we introduce another coefficient

Finally, we list the computation time cost by these two numerical methods in Table

From the comparison performed above, we can see that the infinity norm of absolute errors

(a) Comparisons of the exact solutions and LB numerical solutions for

Comparison of the infinity norm of absolute errors

Our model | Implicit difference scheme equation ( | |
---|---|---|

6.731748 × | 5.892235 × | |

3.288136 × | 3.013750 × | |

1.615917 × | 1.529798 × | |

7.978385 × | 7.727516 × |

The convergence rate

Our model | Implicit difference scheme equation ( | |
---|---|---|

2.047284 | 1.955117 | |

2.034842 | 1.970031 | |

2.025369 | 1.979676 |

The parameter

Our model | Implicit difference scheme equation ( | |
---|---|---|

1.346350 | 0.1178447 | |

1.315254 | 0.1205500 | |

1.292734 | 0.1223838 | |

1.276542 | 0.1236403 |

The comparison of the time cost of the LB model presented in this paper and the implicit difference in Example 1. The parameters for the LB model are the same as those in Table

Our model (s) | Implicit difference scheme equation ( | |
---|---|---|

2.5s | 5.3s |

In this example, the following FDAE is simulated:

The parameters are

The initial condition is given by

The Dirichlet boundary conditions are given by the following equations:

The exact solution to this problem is given as follows:

From equations (

By employing equation (

The numerical results are shown in Figure

Figures

The relationship between the errors and spatial positions

(a) Comparisons of the exact and LB numerical results for

Finally, the following form of FDAE is considered:

The parameters are

The initial condition of the problem is given by

The Dirichlet boundary conditions are given by

The exact solution to the problem is

By applying equations (

In Figure

(a) Comparisons of the LB numerical results and exact solutions for

In this paper, an LB model is proposed to investigate the FDAE. The Riemann–Liouville definition and the Grünwald–Letnikov definition are applied to the time-fractional derivatives. Then, 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. The 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. Three numerical examples are carried out to test the validity of our model. The 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. The results demonstrate that the proposed model and the implicit finite difference scheme have the same convergence rate and convergence order. The 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. The 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

In this section, the detailed recovery process of the FDAE is provided. First, we take equation (

Now, based on equation (

Substituting equations (

If we choose the source term

This is the FDAE with second-order accuracy of the truncation error.

From equation (

In equation (

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors would like to dedicate this article to Prof. Guangwu Yan, who unfortunately passed away before this paper was submitted to the journal. The authors will miss him forever! This work was supported by the National Nature Science Foundation of China (grant nos. 11272133 and 11602033).