A-Stability Versus L-Stability

Severe as the requirement of A-stability is, in one sense it is not severe enough. Consider, for example, the application of the Trapezoidal rule to the scalar test equation a complex constant with . We obtain

Since the Trapezoidal rule is A-stable, , as for all fixed .

However,

and if and is not small then will be close to . Thus will decay to zero only very slowly, and it follows that an A-stable method may be unsatisfactory for an excessively stiff system. Contrast the behavior of the method [the backward Euler method) for which

and the RHS tends to zero as and we can expect a rapid decay of even for moderately large .

Definition: (Ehle) A one step method is said to be L-stable if it is A-stable and, in addition, when applied to the scalar test equation a complex constant with it yields , where as . Note that