$$
C(h) \frac{\partial h }{\partial t}-
\frac{\partial }{\partial z}[k\frac{\partial h }{\partial z}]
+\frac{\partial k }{\partial z}=0
$$
where
Using a 1st order Taylor expansion
$$ C_i^{n+1,m} \times \frac{ h_i^{n+1,m}- h_i^n }{\Delta t}- [\frac{ k^{n+1,m}{i+1/2}(h{i+1}^{n+1,m+1}-h_i^{n+1,m+1}) -k^{n+1,m}{i-1/2}(h{i}^{n+1,m+1}-h_{i-1}^{n+1,m+1})} { \Delta z^2 }] \ +\frac{k^{n+1,m}{i+1/2}-k^{n+1,m}{i-1/2}}{\Delta z} $$
$$ w_{i-1}h_{i-1}+w_{i}h_{i}+w_{i-1}h_{i-1}=b_i $$ where the weight could be calculated from above equation easily.
we now have linear equation to be solve at each iteration m
Define a picking up matrix
we have
$$
PAh=b-\hat{P}Ah
$$
Sine the zero column of