These methods constitute a broad class of techniques for numerically solving differential equations. The methods have three distinguishing features: (1) they are called one-step methods as they require information at the preceding point only; (2) they agree with the Taylor series (Eqn [1.24]) through to order p, which is therefore called the order of the method; and (3) they do not require the evaluation of any higherorder derivatives as they use only the given first-derivative expression, f. This latter property makes the methods more useful than the Taylor series method. However, the price paid for not evaluating the higher-order derivatives is to evaluate far more than one value beyond (tm, ym), when stepping from (tm, ym) to (tm+1, ym+1).
A geometrical representation is useful for understanding these methods, but remember that algebraic analysis is still needed for their ultimate verification. Suppose that the solution ym at the point t = tm is known. Then a line is drawn through the point (tm, ym) such that its slope is ym' = f[tm,ym]. See Fig. 1.9 where the heavy line denotes the exact but unknown solution and the line just described is denoted by L1 . Then we let ym+1 be the point where L1 intersects the ordinate erected at t = tm+1 = tm+ h. Hence, the equation for L1 is y = ym + ym' (t - tm) , [1.29]
The error at t = tm+1 is shown in Fig. 1.9 as e, and Eqn [1.32] agrees with the Taylor series through to terms in h, so the truncation error is represented by eT = K h2 . [1.33]
Note that although the point (tm, ym) in Fig. 1.10 is drawn lying on the exact (unknown) curve, in practice it is an approximation, so it will not necessarily do so. Equation (1.31) is called Euler's point-slope method and is the oldest and best known numerical method for integrating a differential equation. However, besides having a relatively large truncation error, it can also be quite unstable. In other words, a small error due to numerical round-off, truncation, or inherent in the underlying function f becomes magnified as t increases. Therefore, more accurate methods have been developed.
In passing, note that according to the definition above, Euler's point-slope method is a first-order Runge-Kutta one, since it agrees with the Taylor series representation of the function through to terms in h.
Euler's point-slope method can be improved in a number of different ways. The two most important ones are called the Improved Euler method and the Modified Euler method and they are discussed next.
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