Fourth-order alternating direction implicit difference scheme to simulate the space-time Riesz tempered fractional diffusion equation

Abstract

The current paper proposes a new high-order finite difference scheme with low computational complexity to solve the space-time fractional tempered diffusion equation. At the first stage, the time derivative has been approximated by a difference scheme with second-order accuracy. Furthermore, in the next step, a compact operator has been employed to discretize the space derivative with fourth-order accuracy. After deriving the time-discrete scheme, its stability is analysed. So, a suitable term is added to the main difference scheme. By adding this term, we could construct the main ADI scheme. In the final stage, the convergence order of the full-discrete scheme based upon the ADI formulation is proved. The convergence order of the constructed technique is $\mathcal{O}((h_x^{\alpha }{)}^4+(h_y^{\beta }{)}^4+{\tau }^2)$. The numerical results show the efficiency of the new technique.

Keywords:

• Tempered fractional PDEs
• space-time fractional diffusion equation
• alternating direction implicit (ADI) method
• unconditional stability and convergence analysis
• finite difference method

2010 Mathematics Subject Classifications:

• 65L60
• 65L20
• 65M70

Acknowledgments

We would like to give our sincere gratitude to reviewers for their comments and suggestions that greatly improved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.