Polynomial Spectral collocation Method for Space Fractional Advection-Diffusion Equation

TL;DR

This paper develops a spectral collocation method for solving space fractional advection-diffusion equations, deriving differentiation matrices, establishing stability, and demonstrating exponential convergence through numerical examples and physical simulations.

Contribution

It introduces a novel spectral collocation approach for space fractional PDEs, including derivation of differentiation matrices and stability analysis.

Findings

Method achieves exponential convergence.

Numerical schemes are stable for various boundary conditions.

Physical simulations validate the method's effectiveness.

Abstract

This paper discusses the spectral collocation method for numerically solving nonlocal problems: one dimensional space fractional advection-diffusion equation; and two dimensional linear/nonlinear space fractional advection-diffusion equation. The differentiation matrixes of the left and right Riemann-Liouville and Caputo fractional derivatives are derived for any collocation points within any given interval. The stabilities of the one dimensional semi-discrete and full-discrete schemes are theoretically established. Several numerical examples with different boundary conditions are computed to testify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for L\'evy-Feller advection-diffusion equation are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases.