Forward and Backward Euler Methods
From (8), it is evident that an error is induced at every time-step due to the truncation of the Taylor series, this is referred to as the local truncation error (LTE) of the method. For the forward Euler method, the LTE is O(h 2 ). Hence, the method is referred to as a first order technique. In general, a method with O(h k +1 ) LTE is said to be of k th order. Evidently, higher order techniques provide lower LTE for the same step size. The truncation error is different from the global error g n . which is defined as the absolute value of the difference between the true solution and the computed solution, i.e. g n = |y e (t n )-y n |. In most cases, we do not know the exact solution and hence the global error is not possible to be evaluated. However, if we neglect roundoff errors, it is reasonable to assume that the global error at the n th time step is n times the LTE, since n is proportional to 1/h. g n should be proportional to LTE /h. This implies that for a k th
order method, the global error scales as h k.
A convergent numerical method is the one where the numerically computed solution approaches the exact solution as the step size approaches 0. Once again, if the true solution is not known a priori, we can choose, depending on the precision required, the solution obtained with a sufficiently small time step as the 'exact' solution to study the convergence characteristics.
Another important observation regarding the forward Euler method is that it is an explicit method, i.e. y n +1 is given explicitly in terms of known quantities such as y n and f (y n ,t n ). Explicit methods are very easy to implement, however, the drawback arises from the limitations on the time step size to ensure numerical stability. In order to see this better, let's examine a linear IVP, given by dy /dt = -ay. y (0)=1 with a >0. As we know, the exact solution
, which is a stable and a very smooth solution with y e (0) = 1 and
. Now, what is the discrete equation obtained by applying the forward Euler method to this IVP? Using Eq. 7, we get