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45 STABILITY In Section 4.4 we investigated some of the features of consistent and conver- gent finite difference approximations as they related to the model parabolic Stability PDE Now we consider the stability of these approximations.This feature of numerical analysis has basically nothing to do with the PDE but rather concerns the unstable growth or stable decay of errors in the arithmetic operations needed to solve the finite difference equations themselves.Although there is a theoretically exact solution to any one of the finite difference approximations, because of round-off error in the computer, errors are com mitted as the explicit calculation is carried out.If an approximation is stable, then we are assured that computational errors can, in principle, be made arbitrarily small.Whether these errors amplify or decay characterizes the stability property of the scheme.

45 STABILITY In Section 4.4 we investigated some of the features of consistent and conver- gent finite difference approximations as they related to the model parabolic Stability PDE Now we consider the stability of these approximations. This feature of numerical analysis has basically nothing to do with the PDE but rather concerns the unstable growth or stable decay of errors in the arithmetic operations needed to solve the finite difference equations themselves. Although there is a theoretically exact solution to any one of the finite difference approximations, because of round-off error in the computer, errors are com mitted as the explicit calculation is carried out. Whether these errors amplify or decay characterizes the stability property of the scheme. If an approximation is stable, then we are assured that computational errors can, in principle, be made arbitrarily small. Once again, we encounter the terms "conditional" and "unconditional," but now they refer to the stability of a particular finite difference approximation. As before, a conditionally stable scheme implies that there are restrictive bounds on the mesh widths, k, k...., above which the numerical solution is unstable and below which the solution is stable. An unconditionally stable algorithm has no such bounds. Of course, an unconditionally unstable algo- rithm has bounds of zero, and no meaningful computation can be carried out with this algorithm. We find, for example, that most explicit finite difference approximations are conditionally stable, whereas most implicit approximations are unconditionally stable. However, this is not always true and many workers have developed unconditionally stable explicit algorithms. The DuFort-Frankel approxima tion, (43.13), is a classic example in which the bound on h, k..... is p