Third order quasi-compact schemes for space tempered fractional diffusion equations

TL;DR

This paper develops third-order quasi-compact numerical schemes for space tempered fractional diffusion equations, providing stability, convergence proofs, and numerical validation for modeling complex diffusion processes.

Contribution

It introduces novel third-order quasi-compact schemes specifically designed for space tempered fractional diffusion equations, differing from schemes for pure fractional derivatives.

Findings

Schemes are proven to be stable and convergent.

Numerical simulations confirm high accuracy and effectiveness.

The methods effectively model transitions among different diffusion types.

Abstract

Power-law probability density function (PDF) plays a key role in both subdiffusion and L\'{e}vy flights. However, sometimes because of the finite of the lifespan of the particles or the boundedness of the physical space, tempered power-law PDF seems to be a more physical choice and then the tempered fractional operators appear; in fact, the tempered fractional operators can also characterize the transitions among subdiffusion, normal diffusion, and L\'{e}vy flights. This paper focuses on the quasi-compact schemes for space tempered fractional diffusion equations, being much different from the ones for pure fractional derivatives. By using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Some numerical simulations are performed to testify the effectiveness and numerical accuracy of the obtained schemes.