Second-order Stable Finite Difference Schemes for the Time-fractional Diffusion-wave Equation

TL;DR

This paper introduces two stable and one conditionally stable second-order finite difference schemes for the time-fractional diffusion-wave equation, enhancing numerical stability and accuracy over existing methods.

Contribution

The paper develops new second-order finite difference schemes with proven stability for the time-fractional diffusion-wave equation, including extensions to advection-reaction terms.

Findings

The first and third schemes are unconditionally stable.

Numerical results confirm the schemes' higher accuracy and stability.

Proposed schemes outperform existing methods in tests.

Abstract

We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. We use the generalized Newton-Gregory formula in time for the second scheme and its modification for the third scheme. While the second scheme is conditionally stable, the first and the third schemes are stable. We apply the methodology to the considered equation with also linear advection-reaction terms and also obtain second-order schemes both in time and space. Numerical examples with comparisons among the proposed schemes and the existing ones verify the theoretical analysis and show that the present schemes exhibit better performances than the known ones.