Stability of Explicit Scheme

Note that the steady value may also be obtained when the lefthand side of Eq. (4.50) is set to zero. From the ratio of Eqs. (4.50) and (4.54) 

The reason for the selection of 5/11 and 5/9 for F in Figs. 4.21 and 4.22 now becomes dear. That is, the case of F 1/2, which causes instability in the difference equations, violates the physics of the problem by leading to a temperature that exceeds its steady value.1 This stability criterion can also be obtained by requiring that the coefficient of 7j in Eq. (4.51) be positive. 

Consider next the effect of enthalpy flow on the stability at inner nodes. The governing equation now becomes 

After rearranging Eq. (4.59) for 6 +1, the same criterion can again be recovered by setting the coefficient of 0 positive. We learn from this result that the presence of enthalpy flow further restricts the size of the time step for a given spatial discretization. A constant source term, say u jk, however, has no effect on the stability of the numerical 

Finally, Let us consider a boundary node—for example, the boundary of a flat plate—now subject to convective heat transfer to an ambient at temperature T. The discretized balance of thermal energy and the Fourier law of conduction applied to a Ax /2-thick boundary difference system (Fig. 4.23) yields for 9 = T — To, 

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