High-order numerical algorithms for the time-fractional convection–diffusion equation

Abstract

In this paper, efficient methods are derived for seeking a numerical solution to the time-fractional convection–diffusion equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time-fractional derivative in the Caputo sense with order in $(0,1)$ is discretized by the $L2$-$1_{\sigma }$ methods with uniform and non-uniform meshes and the spatial derivative is approximated by the local discontinuous Galerkin (LDG) finite element methods. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm.

Keywords:

• Caputo derivative
• local discontinuous Galerkin method
• non-uniform time mesh
• stability
• convergence

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 65M60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant number 12101266].