Efficient numerical algorithms for three-dimensional fractional partial differential equations

TL;DR

This paper develops efficient, unconditionally stable second-order numerical algorithms for three-dimensional fractional PDEs, including advection-diffusion and Riesz diffusion equations, with theoretical proofs and numerical verification.

Contribution

It introduces a locally one-dimensional method with second-order accuracy for 3D fractional PDEs, improving stability and convergence, especially for Riesz fractional diffusion equations.

Findings

Algorithms are unconditionally stable.

Methods are second-order convergent in space and time.

Improved scheme reduces splitting error for Riesz fractional diffusion.

Abstract

This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional diffusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.