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

TL;DR

This paper introduces a second-order midpoint approximation formula for the Riemann-Liouville derivative, enabling more accurate numerical solutions for time-fractional differential equations, with proven stability and convergence.

Contribution

The paper proposes a novel second-order approximation scheme for the Riemann-Liouville derivative at midpoints, improving accuracy for fractional PDEs and facilitating high-order numerical methods.

Findings

The new formula achieves ( au^2) accuracy in discretizing the Riemann-Liouville derivative.

The scheme is applied to time fractional Cable equations in 1D and 2D.

Numerical experiments confirm stability and convergence of the proposed methods.

Abstract

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.