Simple ECEM algorithms using function values only

In this paper, we improve the error corrected Euler method(ECEM) introduced in [11] by evaluating function values only at local nodes in each time interval. As a result, one can avoid computations of Jacobian matrices on each time interval so that the algorithms become simpler to implement in solving various class of time dependent differential equations numerically. The proposed ECEM formula resembles to the Runge-Kutta method in its representations but both methods have different characteristic properties.