# Multiple-Relaxation-Time Lattice Boltzmann scheme for Fractional Advection-Diffusion Equation

## TL;DR

This paper develops a lattice Boltzmann method with multiple relaxation times for solving higher-dimensional fractional advection-diffusion equations, improving accuracy and stability over traditional BGK schemes.

## Contribution

It introduces an MRT-based lattice Boltzmann scheme for fractional PDEs in higher dimensions, enhancing accuracy and stability in complex anisotropic and flow conditions.

## Key findings

- MRT scheme accurately approximates fractional diffusion in 2D and 3D.
- MRT outperforms BGK in anisotropic and flow destabilized scenarios.
- Solutions validated against random walks and exact solutions.

## Abstract

Partial differential equations (p.d.e) equipped of spatial derivatives of fractional order capture anomalous transport behaviors observed in diverse fields of Science. A number of numerical methods approximate their solutions in dimension one. Focusing our effort on such p.d.e. in higher dimension with Dirichlet boundary conditions, we present an approximation based on Lattice Boltzmann Method with Bhatnagar-Gross-Krook (BGK) or Multiple-Relaxation-Time (MRT) collision operators. First, an equilibrium distribution function is defined for simulating space-fractional diffusion equations in dimensions 2 and 3. Then, we check the accuracy of the solutions by comparing with i) random walks derived from stable L\'evy motion, and ii) exact solutions. Because of its additional freedom degrees, the MRT collision operator provides accurate approximations to space-fractional advection-diffusion equations, even in the cases which the BGK fails to represent because of anisotropic diffusion tensor or of flow rate destabilizing the BGK LBM scheme.