ABSTRACT

The Euler method developed in an earlier chapter is a good starting point for those learning about numerical methods for differential equations, but is a poor ending point for those computing numerical solutions to initial-value problems arising in applications. Accordingly, this chapter is devoted to expanding the reader’s repertoire of numerical methods for first-order initial-value problems. While a number of methods are discussed, most of the chapter is devoted to the “improved Euler method” and the “classic Runge-Kutta method”, both of which typically deliver much more accurate approximations, and both of which are used much more often in real-world applications than the Euler method. After discussing the weakness inherent in the basic Euler method, the chapter turns to developing the improved Euler method and the classic Runge-Kutta method and then illustrating their use by example. Following that is the discussion of the accuracy of each, with the accuracy of the three methods (basic Euler, improved Euler and Runge-Kutta) compared both through their theoretical error bounds and by a graphical comparison of solutions generated by the three methods.