In this paper, we derive a class of exponential methods for solving th
e stiff initial-value problems. The present methods are unconditionall
y stable and satisfy a discrete maximum principle. These include an ev
en order of accuracy when the perturbation parameter, epsilon, is fixe
d and have the property that if epsilon is of order h they reduce to f
irst order accuracy. Also, these methods are optimal when epsilon -->
0. Finally, good results and comparison with the uniform second-order
scheme are considered.