Lattice Boltzmann method for the fractional advection-diffusion equation

Area of Science:

• Fluid Dynamics
• Computational Science
• Environmental Engineering

Background:

• Mass transport phenomena, like phosphorus in soils and solutes in rivers, are crucial in science and engineering.
• Classical advection-diffusion equations (ADE) fail to describe certain diffusion behaviors, termed abnormal or superdiffusion.
• Fractional advection-diffusion equations (FADE) accurately model superdiffusion but are computationally challenging.

Purpose of the Study:

• To develop a novel and efficient numerical method for solving the fractional advection-diffusion equation (FADE).
• To introduce the lattice Boltzmann method (LabFADE) as a simplified and effective approach for FADE solutions.
• To validate the accuracy and applicability of the LabFADE method for complex mass transport problems.

Main Methods:

• A new lattice Boltzmann method (LabFADE) was developed by transforming the FADE into a form solvable by the lattice Boltzmann method.
• The LabFADE method was validated using benchmark simulations, including point-source diffusion, steady-state boundary value problems, and unsteady diffusion with source/sink terms.
• The accuracy and convergence of LabFADE were assessed by analyzing the effects of skewness (β), fractional order (α), and relaxation time (τ), comparing results with analytical solutions.

Main Results:

• The LabFADE method successfully solves the FADE, offering advantages over existing complex numerical techniques.
• Simulations confirmed the method's ability to handle various diffusion scenarios, including those with source and sink terms.
• Numerical predictions demonstrated second-order accuracy, aligning well with analytical solutions.

Conclusions:

• The developed LabFADE method provides an efficient and accurate solution for the fractional advection-diffusion equation.
• This novel approach simplifies the complex solution procedures typically associated with FADE.
• The LabFADE method enhances the potential for wider application of FADE in studying intricate mass transport processes across scientific and engineering disciplines.