A new second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and its application

Compared to the the classical first-order Gr\"unwald-Letnikov formula at time $t_{k+1} (\textmd{or}\, t_{k})$, we firstly propose a second-order numerical approximate scheme for discretizing the Riemann-Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank-Niclson technique applied to the time-fractional differential equations. The established formula has the following form $$\begin{array}{lll} \displaystyle \,_{\mathrm{RL}}{{{\mathrm{D}}}}_{0,t}^{\alpha}u\left(t\right)\left|\right._{t=t_{k+\frac{1}{2}}}= \tau^{-\alpha}\sum\limits_{\ell=0}^{k} \varpi_{\ell}^{(\alpha)}u\left(t_k-\ell\tau\right) +\mathcal{O}(\tau^2),\,\,k=0,1,\ldots, \alpha\in(0,1), \end{array}$$ where the coefficients $\varpi_{\ell}^{(\alpha)}$ $(\ell=0,1,\ldots,k)$ can be determined via the following generating function $$\begin{array}{lll} \displaystyle G(z)=\left(\frac{3\alpha+1}{2\alpha}-\frac{2\alpha+1}{\alpha}z+\frac{\alpha+1}{2\alpha}z^2\right)^{\alpha},\;|z|<1. \end{array}$$ Applying this formula to the time fractional Cable equations with Riemann-liouville derivative in one or two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.