A new difference scheme for the time fractional diffusion equation

TL;DR

This paper introduces a new difference scheme for the time fractional diffusion equation based on a novel fractional derivative approximation, achieving higher order accuracy and proven stability and convergence.

Contribution

It presents a new difference analog of the Caputo fractional derivative and develops higher-order schemes with proven stability and convergence for variable coefficient fractional diffusion equations.

Findings

The new scheme achieves second and fourth order spatial accuracy.

Stability and convergence are rigorously proved.

Numerical tests confirm theoretical results.

Abstract

In this paper we construct a new difference analog of the Caputo fractional derivative (called the $L2$-$1_\sigma$ formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and forth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid $L_2$ - norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by the numerical calculations carried out for some test problems.