1.2. Euler’s Method 17 necessary to fall back on constructing an approximate solution nu- merically with the aid of a computer. But what algorithm should we use to program the computer? A natural first guess is succes- sive approximations. But while that is a powerful theoretical tool for studying the general properties of initial value problems (and in particular for proving existence and uniqueness), it does not lead to an efficient algorithm for constructing numerical solutions. In fact there is no one simple answer to the question of what numerical algorithm to use for solving ODEs, for there is no single method that is “best” in all situations. While there are integration routines (such as the popular fourth-order Runge-Kutta integration) that are fast and accurate when used with many of the equations one meets, there are many situations that require a more sophisticated approach. Indeed, this is still an active area of research, and there are literally dozens of books on the subject. Later, in the chapter on numerical methods, we will introduce you to many of the subtleties of this topic, but here we only want to give you a quick first impression by describing one of the oldest numerical approaches to solving an initial value problem, the so-called “Euler Method”. While rarely an optimal choice, it is intuitive, simple, and effective for some purposes. It is also the prototype for the design and analysis of more sophisti- cated algorithms. This makes it an excellent place to become familiar with the basic concepts that enter into the numerical integration of ODE. In what follows we will suppose that f(t, y) is a C1 time-depend- ent vector field on Rd, to in R and yo in Rd. We will denote by σ(f, yo,to,t) the solution operator taking this data to the values y(t) of the maximal solution of the associated initial value problem. By definition, y(t) is the function defined on a maximal interval I = [to,to+T∗), with 0 T∗ ∞, satisfying the differential equation dy dt = f(t, y) and the initial condition y(to) = yo. The goal in the numerical integration of ODE is to devise effective methods for approximating such a solution y(t) on an interval I = [to,to + T ] for T T∗. The strategy that many methods use is to discretize the interval I using N + 1 equally spaced gridpoints tn := to + nh, n = 0,... , N with h = T N so that t0 = to and tN = to + T and then use some algorithm
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