I am reading a paper by Zhang 2007 about reactive chemical transport where a fully implicit method was used in solving the transport equation given by,
"At $\left(n+1\right)$th time step, equation is approximated by
$$A\frac{\left(E_{i}\right)^{n+1}-\left(E_{i}\right)^{n}}{\Delta t}+\frac{\partial A}{\partial t}E_{n}+\frac{\partial\left(QE_{i}^{m}\right)}{\partial x}-\frac{\partial}{\partial x}\left[K_{x}A\frac{\partial E_{i}^{m}}{\partial x}\right]=M_{E_{i}}^{as}+M_{E_{i}}^{rs}+M_{E_{i}}^{os1}+M_{E_{i}}^{os2}+M_{E_{i}}^{is}+AR_{E_{i}}$$ According to fully-implicit scheme, equation can be separated into two equations as follows $$A\frac{\left(E_{i}\right)^{n+\frac{1}{2}}-\left(E_{i}\right)^{n}}{\Delta t}+\frac{\partial A}{\partial t}E_{n}+\frac{\partial\left(QE_{i}^{m}\right)}{\partial x}-\frac{\partial}{\partial x}\left[K_{x}A\frac{\partial E_{i}^{m}}{\partial x}\right] =M_{E_{i}}^{as}+M_{E_{i}}^{rs}+M_{E_{i}}^{os1}+M_{E_{i}}^{os2}+M_{E_{i}}^{is}+AR_{E_{i}}$$ $$\frac{\left(E_{i}\right)^{n+1}-\left(E_{i}\right)^{n+\frac{1}{2}}}{\Delta t} =0$$" How does the equation separates using the fully implicit scheme? Any ideas will be accepted.