Solving thermal conduction problem involves discretizing equations like

\frac{\partial T}{\partial t} = \nabla \cdot \left ( T^n \nabla T \right )

Using the first equation, we would write

\frac{\partial T}{\partial t} = \nabla \cdot \left ( T_{m+1}^n \nabla T_{m+1} \right )

which we could expand using the identities

T^n_{m+1} \approx T^n_m+n T^{n-1}_m \left ( T_{m+1}-T_m \right ) = nT^{n-1}_m T_{m+1} + (1-n) T^n_m

and

\nabla T^n_{m} = n T^{n-1}_{m} \nabla T_{m}

\frac{\partial T}{\partial t} = \nabla \cdot \left [ \left ( nT^{n-1}_m T_{m+1} + (1-n) T^n_m \right ) \nabla T_{m+1} \right ]

\frac{\partial T}{\partial t} = n(n-1) T_m^{n-2} T_{m+1} \nabla T_m \cdot \nabla T_{m+1} + n T_m^{n-1} \nabla T_{m+1} \cdot \nabla T_{m+1} + n(1-n) T^{n-1}_m \nabla T_m \cdot \nabla T_{m+1} + \left ( nT^{n-1}_m T_{m+1} + (1-n) T^n_m \right ) \nabla^2 T_{m+1}

which can be simplified

\frac{\partial T}{\partial t} = n(n-1) T_m^{n-1} \left ( T_{m+1} - T_m \right ) \nabla T_m \cdot \nabla T_{m+1} + n T_m^{n-1} \nabla T_{m+1} \cdot \nabla T_{m+1} + \left ( nT^{n-1}_m T_{m+1} + (1-n) T^n_m \right ) \nabla^2 T_{m+1}

We could also start with

\frac{\partial T}{\partial t} = \frac{1}{n+1}\nabla^2 T_{m+1}^{n+1} \approx \frac{1}{n+1}\nabla^2 \left [ nT_m^{n}T_{m+1} + (1-n) T^{n+1}_{m} \right ]

which can be expanded as

\frac{\partial T}{\partial t} = \frac{1}{n+1} \nabla \left [ n^2 T^{n-1}_m \nabla T_m T_{m+1} + n T^n_m \nabla T_{m+1} + (1-n)(n+1)T^n_m \nabla T_m \right ]

\frac{\partial T}{\partial t} = \frac{1}{n+1} \left [n^2 (n-1) T^{n-2}_m \nabla T_m \nabla T_m T_{m+1} + n^2 T^{n-1}_m \nabla^2 T_m T_{m+1} + n^2 T^{n-1}_m \nabla T_m \nabla T_{m+1} + n^2 T^{n-1}_m \nabla T_m \nabla T_{m+1} + nT^n_m \nabla^2 T_{m+1} + (1-n)(n+1)nT^{n-1}_m \nabla T_m \nabla T_m + (1-n)(n+1)T^n_m \nabla^2 T_m \right ]

\frac{\partial T}{\partial t} = \frac{1}{n+1} \left [n^2 (n-1) T^{n-2}_m \nabla T_m \nabla T_m T_{m+1} + 2n^2 T^{n-1}_m \nabla^2 T_m T_{m+1} + n^2 T^{n-1}_m \nabla T_m \nabla T_{m+1} + nT^n_m \nabla^2 T_{m+1} + (1-n)(n+1)nT^{n-1}_m \nabla T_m \nabla T_m + (1-n)(n+1)T^n_m \nabla^2 T_m \right ]