. J. Eyre in \cite{{D. J. Eyre(1998)}} to craft energy stable numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. The basic idea is to treat the convex part implicitly while to treat the concave part explicitly. Then a {\bf  semi-discrete  scheme} for (\ref{equ:crp}) is proposed as follows
\begin{equation}\label{equ:num}
 \frac{f_{0}(X)}{x^{n}(1-x^{n})}\frac{x^{n+1}-x^{n}}{\tau}=
-\frac{\partial}{\partial X}\left(\frac{f_{0}(X)}{\frac{\partial x^{n+1}}{\partial X}}\right)-f_{0}(X)\frac{1-2x^{n}}{x^{n}(1-x^{n})},
\end{equation}
where $\tau$ is the time step and $x^{n}=x(X,t^{n})$ is the solution at time $t^{n}=n\tau$, $n\in\mathbb{N}^{+}$.

\begin{rem}
   \eqref{equ:num} is also a Variational Particle Scheme. We explain the fact in the framework of optimal transport theory. Let $\Omega=[0,1]$. We denote by $\mathscr{P}(\Omega)$ the space of $\mathfrak{L}^{1}$ measure on $\Omega$, non-negative functions with unit integral and finite second moments, where $\mathfrak{L}^{1}$ is the Lebesgue measure. $f^{n}\in\mathscr{P}(\Omega)$ is the approximation to solution of equation \eqref{eqcm}-\eqref{eqwf} at time $t_{n} = n\tau$, $n\in\mathbb{N}$. We fix a reference density $f^{0}$ and consider a time-dependent family of transport maps $x(\cdot,t^{n}):[0,1]\rightarrow [0,1]$ such that $x(\cdot,t^{n})\#f^{0}=f ^{n}\mathfrak{L}^{1}$ for all $n\in\mathbb{N}^{+}$, where $\#$ denotes the push-forward of measures.

Then the map from $x^n$ to $x^{n+1}$ is an optimal transport in the sense that $x^{n+1}$ is the minimizer of
  the cost functional
$$F(x) :=\int_{0}^{1}\frac{1}{2\tau}\frac{f_{0}(X)}{x^{n}(1-x^{n})}|x-x^{n}|^{2}+
f_{0}(X)\ln\left(\frac{f_{0}(X)}{\frac{\partial x}{\partial X}}\right)+ f_{0}(X)\frac{1-2x^{n}}{x^{n}(1-x^{n})}xdX.$$
Some relevant descriptions on optimal transport can be found in \cite{M. Westdickenberg(2010)}.
\end{rem}

\subsection{The fully discrete scheme}
We begin with the definition of inner-product, difference operators and summation-by-parts in one dimension.
Let $h=\frac{1}{N}$, $N\in\mathbb{N}^{+}$ be the spatial step. Denote by $X_{r}=X(r)=r h$, where $r$ takes on integer and half integer values.  Let $\mathcal{E}_{N}$ and $\mathcal{C}_{N}$ be the spaces of functions whose domains are $\{X_{i}\ |\ i=0,...,N\}$ and $\{X_{i-\frac{1}{2}}\ |\ i=1,...,N\}$ respectively. In component form, these functions are identified via
$l_{i}=l(X_{i})$, $i=0,...,N$, for $l\in\mathcal{E}_{N}$, and $\phi_{i-\frac{1}{2}}=\phi(X_{i-\frac{1}{2}})$,  $i=1,...,N$, for $\phi\in\mathcal{C}_{N}$.

\indent{Let} $l$, $g\in\mathcal{E}_{N}$ and $\phi$, $\psi\in\mathcal{C}_{N}$. We define the ``inner-product" on space $\mathcal{E}_{N}$ and $\mathcal{C}_{N}$  respectively as
\begin{equation}\label{def:inner-integer}
[l\big|g]=h \sum\limits_{i=1}^{N-1} l_{i} g_{i},
\end{equation}
\begin{equation}\label{def:inner-half} (\phi\big|\psi)=h\sum\limits_{i=1}^{N}\phi_{i-\frac{1}{2}}\psi_{i-\frac{1}{2}}.
\end{equation}
\indent{The} difference operator $D_{h}:\mathcal{E}_{N}\rightarrow\mathcal{C}_{N}$ and $d_{h}:\mathcal{C}_{N}\rightarrow\mathcal{E}_{N}$, and the average operator $A:\mathcal{E}_{N}\rightarrow\mathcal{C}_{N}$  can be defined as respectively as
\begin{align}\label{equ:dif1}
& (D_{h}l)_{i-\frac{1}{2}}= (l_{i}-l_{i-1})/h,\ i=1,...,N, \\
&\label{equ:dif2}  (d_{h}\phi)_{i}= (\phi_{i+\frac{1}{2}}-\phi_{i-\frac{1}{2}})/h,\ i=1,...,N-1, \\
&\label{equ:ave1} (Al)_{i-\frac{1}{2}}= (l_{i}+l_{i-1})/2,\ i=1,...,N.
\end{align}
Then we have the following result of summation-by-parts.
\begin{lem}
Let $\phi\in\mathcal{C}_{{N}}$ and $l\in\mathcal{E}_{N}$. Then
$(D_{h}l\big|\phi)=-[l \big|d_{h}\phi]+l_{N}\phi_{N-\frac{1}{2}}-l_{0}\phi_{\frac{1}{2}}$.
\label{lemmasum}
\end{lem}
Let $\textbf{Q}:=\{l \in\mathcal{E}_{N}\ |\ l_{i-1}<l_{i},\ 1\leq i\leq N;\ l_{0}=0,\ l_{N}=1\}$ and its boundary set $\partial\textbf{Q}:=\{l \in\mathcal{E}_{N}\ |\ l_{i-1}\leq l_i,\ 1\leq i\leq N,\ and\  l_{i}=l_{i-1},\ for\ some\ 1\leq i\leq N; l_{0}=0,\ l_{N}=1\}$. Then $\bar{\textbf{Q}}:=\textbf{Q}\cup\partial\textbf{Q}$ is a closed convex set.

The {\bf  fully discrete scheme} is formulated as follows: Given $x^{n} \in\textbf{Q}$, find $x^{n+1}=(x^{n+1}_{0},...,x^{n+1}_{N})\in\textbf{Q}$ such that


\begin{equation}\label{equ:numnum}
 \frac{f_{0}(X_{i})}{x^{n}_{i}(1-x^{n}_{i})}\frac{x^{n+1}_{i}-x^{n}_{i}}{\tau}=
-d_{h}\left(\frac{Af_{0}(X)}{D_{h}x^{n+1}}\right)_{i}
-f_{0}(X_{i})\frac{1-2x^{n}_{i}}{x^{n}_{i}(1-x^{n}_{i})},\ 1\leq i\leq N-1.
\end{equation}
 (\ref{equ:numnum}) is still a nonlinear system. Newton's iteration method can be applied to solve it.

\noindent{\bf Damped Newton's iteration.} \ \ Set $x^{n+1, 0}= x^n$. For $k=0,1,2,\cdots, x^{n+1,k+1}=x^{n+1,k} + \omega(\lambda)\delta_x$ such that
\begin{align}\label{equ:numnum1}
\frac{f_{0}(X_{i})}{x^{n}_{i}(1-x^{n}_{i})}\frac{ \delta_{x_{i}}}{\tau} &- d_h \left(\frac{Af_{0}(X)}{(D_{h}x^{n+1,k})^2} D_h \delta_{x_i}\right)_{i}= -
 \frac{f_{0}(X_{i})}{x^{n}_{i}(1-x^{n}_{i})}\frac{x^{n+1,k}_{i} -x^{n}_{i}}{\tau} \notag \\
&
-d_{h}\left(\frac{Af_{0}(X)}{D_{h}x^{n+1,k}}\right)_{i}
-f_{0}(X