A high-order unconditionally stable numerical method for a class of multi-term time-fractional diffusion equation arising in the solute transport models

Abstract

In this paper, we study a high-order unconditionally stable numerical method to approximate the class of multi-term time-fractional diffusion equations. This type of problem appears in the modelling of transport of certain quantities such as heat, mass, energy, solutes in ground water and soils. The multi-term time-fractional derivative is approximated by using the Crank–Nicolson method for the Caputo's time derivative. The space derivative is approximated by using the collocation method based on quintic B-spline basis functions. We have established the stability and convergence analysis of the proposed numerical scheme thoroughly, and it is shown that the order of convergence in space variable is almost four and in the time variable is $\mathcal{O}\left(\mathrm{\Delta }{t}^{2-max\{\gamma ,{\gamma }_i\}}\right)$. To prove the accuracy and efficiency of the developed method, we consider four numerical examples and perform the numerical simulation. The developed algorithm works well andvalidate the theoretical results. The developed method is fourth-order convergent in the space variable, which is almost two orders of magnitude higher than the other spline collocation methods.

Keywords:

• Multi-term time fractional diffusion equations
• Crank–Nicolson method for the Caputo derivative
• Quintic B-spline basis functions
• stability and convergence analysis

2000 Mathematics Subject Classifications:

• 65M99
• 65N35
• 65N55
• 65L10
• 65L60
• 34B16

Disclosure statement

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