Three-Term Taylor Series(a) Replace the so-called two-term Taylor estimate (equation (8) i...

Three-Term Taylor Series

(a) Replace the so-called two-term Taylor estimate (equation (8) in Problem 23) by the three-term result:

Compute the second derivative y″ in terms of ft and fy (partial derivatives of f, with respect to t and y, respectively), and deduce the three-term Taylor approximation:

(b) Show that the local discretization error en+1 in this scheme is.

(c) Apply the method to the IVP in Problem 1 and compare the results.

(d) Repeat (c) for Problem 2.

Richardson’s Extrapolation Euler’s method gives first-order approximations, but can readily be used to make more accurate higher-order approximations. The basic idea starts with equation (7).

When Euler’s method starts at t0, the accumulated discretization error in the approximation at t* = t0 + nh is bounded by a constant times the step size h, as shown in equation (7). Thus, at t* the true solution can be written

where yn is the first-order Euler approximation after n steps. Now repeat the Euler computations with step size h/2, so that 2n steps are needed to reach t*. Equation (9)becomes

Subtracting (9) from two times (10) eliminates the Ch term, thus giving a second-order approximation

This technique of raising the order of an approximation by using both yn and the half-step approximation y2n is called Richardson’s extrapolation; it can be used with any numerical DE method.