Introduction to Numerical Methods for Time Dependent Differential Equations

CHAPTER 3

HIGHER-ORDER METHODS

In this chapter we introduce one-step numerical methods that have been developed to compute higher-precision approximations of the exact solutions of ordinary differential equations. We introduce the second-order Taylor method, the improved Euler’s method, and a whole class of methods called Runge-Kutta methods. At the end we generalize the concepts of stability region and truncation error introduced for Euler’s method in Chapter 2.

3.1 Second-order Taylor method

To begin with, consider a differential equation

(3.1)

where F(t) is a smooth function. We want to derive a more accurate approximation than the one obtained from the explicit Euler method. If we think of the grid and the notation for grid functions introduced in Section 2.1, we have, by Taylor expansion,

(3.2)

Now we can use the differential equation to express dy/dt and d²y/dt² in terms of y, F, and dF/dt. This is obvious for dy/dt. Furthermore, the function d²y/dt² is obtained by differentiating the differential equation,

Therefore, we can write (3.2) as

where

Neglecting terms of order (