Equivalence of C0 and C1 methods for ODE's

In this paper, it is proved that exponentially-fitted C0 techniques based on piecewise linearization on two adjacent intervals which employ conditions as the three points in this interval and yield piecewise analytical solutions, also provide piecewise smooth solutions everywhere. The proof is based on the use of exponential C1 methods based on the piecewise linearization of nonlinear ordinary differential equations in two adjacent, nonoverlapping intervals, and the imposition of continuity conditions at the end points of these intervals plus the condition of smoothness at the common boundary point of these adjacent intervals. It is also shown that this smoothness requirement comes at the expense of a loss in accuracy of C1 methods, and that the resulting piecewise smooth method is only first-order accurate for small values of the perturbation parameter.