On the numerical treatment of heat conduction problems with mixed boundary conditions

Abstract:

The two-dimensional problem of heat conduction in a rectangle where the temperature is prescribed over a portion of the boundary while the temperature gradient is prescribed over the remainder of the boundary, may be treated numerically by replacing the differential equation of heat conduction and the equations expressing the given initial and boundary conditions by their difference analogs and solving the resulting system. It is shown that if the scheme is to be stable the intervals $\Delta x$ and $\Delta y$ must be chosen so that $k\Delta t/{(\Delta x)^2} + k\Delta t/{(\Delta y)^2} \leqq \tfrac {1}{2}$.