An implicit Keller Box numerical scheme for the solution of fractional subdiffusion equations

In this work, we present a new implicit numerical scheme for fractional subdiffusion equations. In this approach we use the Keller Box method [1] to spatially discretise the fractional subdiffusion equation and we use a modified L1 scheme (ML1), similar to the L1 scheme originally developed by Oldham and Spanier [2], to approximate the fractional derivative. The stability of the proposed method was investigated by using Von-Neumann stability analysis. We have proved the method is unconditionally stable when $0<{\lambda}_q <\min(\frac{1}{\mu_0},2^\gamma)$ and $0<\gamma \le 1$, and demonstrated the method is also stable numerically in the case $\frac{1}{\mu_0}<{\lambda}_q \le 2^\gamma$ and $\log_3{2} \le \gamma \le 1$. The accuracy and convergence of the scheme was also investigated and found to be of order $O(\Delta t^{1+\gamma})$ in time and $O(\Delta x^2)$ in space. To confirm the accuracy and stability of the proposed method we provide three examples with one including a linear reaction term