Hi, I am working on  phase change modelling using the enthalpy method. The governing equations for phase change is often reported in the following form:

A_s * h_s + A_w * h_w + A_p * h_p  + A_e * h_e + A_n * h_n = Q;

where s, w , n, s and p indicate the location of the node, Q is the source term, h is the enthalpy. The link to a paper that used that equation is: http://www.sciencedirect.com/science/article/pii/0038092X9390029N 

My question is: to solve for each node (p), I need the enthalpy of neighboring nodes (n, s, e ,w). But for the node at the boundary, there are not enough neighboring node, how should I treat the node that at the boundary. Please see the attached PDF for further elaboration on my problem. 

Thanks for helps !!!!

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