Transient two-and three-dimensional temperature fields

We will assume that the material properties are constant. The heat conduction equation for planar, transient temperature fields with heat sources has the form 

The temperature at an intersection point of the planar grid in Fig. 2.48 will be 

The two second derivatives in the x- or y-directions are approximated by the central difference quotient, so that 

Using these difference quotients, we obtain, from (2.298), the explicit finite difference formula 

Equations (2.302) and (2.303) enable us to explicitly calculate the temperatures t U at time = t0 + At from the initial temperature distribution .. These can then be used to calculate the temperature of the next time level etc. This difference method is relatively simple to program. It is also suitable for a temperature dependent thermal power density W(d), because WU is calculated for the already known temperature The disadvantage of this explicit difference method lies in its limited stability, cf. 2.4.1.2. The stability condition, from which none of the coefficients on the right hand side of (2.302) are allowed to be negative, limits the time step 

---