approximations of and at $(x_{i,j},t_n)$ are given by $$\begin{aligned}
D_{\Delta t}^{\alpha} M_{i,j}^{n} = \frac{1}{\rho_\alpha} \left( M_{i,j}^{n} - \sum_{k=0}^{n-1} c_k^{n} M_{i,j}^{k} \right),\label{eq:discr_frac_deriv_forw}\\
\bar D_{\Delta t}^{\alpha} U_{i,j}^{n} = \frac{1}{\rho_\alpha}\left( U_{i,j}^{n} - \sum_{k=n+1}^{N} {\overline c}_{n}^k U_{i,j}^k\right) ,\label{eq:discr_frac_deriv_back}\end{aligned}$$ where $$\rho_\alpha = \Gamma(2-\alpha)\Delta t^{\alpha}$$ and the forward and backward coefficients are defined by $$c_k^n = 
\begin{cases}
n^{1-\alpha}-(n-1)^{1-\alpha}, \quad k=0,\\
2(n-k)^{1-\alpha} - (n+1-k)^{1-\alpha} - (n-1-k)^{1-\alpha}, \quad k=1,\dots, n-1,
\end{cases}$$ and $${\overline c}_n^k = 
\begin{cases}
2(k-n)^{1-\alpha} - (k+1-n)^{1-\alpha} - (k-1-n)^{1-\alpha}, \quad k=n+1,\dots, N-1, \\
(N-n)^{1-\alpha}-(N-1-n)^{1-\alpha}, \quad k=N.
\end{cases}$$ In the following lemma, we summarize some properties of the coefficients $c_k^n$, ${\overline c}_n^k$.

\[lem:c\_properties\] We have

1.  $c_k^{n} >0$ for $0\le k \leq n-1$ and ${\overline c}_n^m >0$ for $ n+1\le m\le N$.

2.  $\bar c_k^n = c_{k+1}^{n+1}$ for $1\le n\le N-1$ and $0\le k\le n-1$.

3.  $\sum_{k=0}^{n-1} c_k^{n}=1$ and $\sum_{k=n+1}^{N} {\overline c}_n^k=1.$

By simple calculation, it can be shown that an equivalent form of the L1 scheme of the discrete forward Caputo derivative is $$\label{discreteCaputo2}
D_{\Delta t}^{\alpha} U_{i,j}^{n} =\sum_{k=0}^nb_k\frac{U_{i,j}^{n-k}-U_{i,j}^{n-1-k}}{\Delta t},$$ where $$b_k=\frac{\Delta t^{1-\alpha}}{\Gamma(2-\alpha)}[(k+1)^{1-\alpha}-k^{1-\alpha}].$$ When written in this form and used for fractional phase field equations it has been shown the L1 scheme of Caputo derivative can preserve on a discrete level maximum principle, energy stability and mass 