A simple finite difference method for plane, steady-state temperature fields

1 A simple finite difference method for plane, steady-state temperature fields 

steady-state temperature fields d = (x,y) with heat sources of thermal power density W are described by the differential equation 

A square grid with mesh size Ax = Ay is chosen for the discretisation, such that 

In the sense of the sign agreement in thermodynamics, the four heat flows will be taken as positive quantities if they flow into the block With that we obtain 

This is an approximate equation as each small, but finite block has only one discrete temperature associated with it and because only the heat conduction between immediate neighbours has been considered. As we can show with the use of the discretisation equations (2.299) and (2.300) for the second derivatives, we also obtain (2.308) by the usual discretisation of the differential equation (2.306). The discretisation error in this case is 0(Ax2) it decreases to zero with the square of the mesh size (= block width). 

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