David F. Griffiths

Long-Term Dynamics

There are many applications where one is concerned with the long-term behaviour of nonlinear ODEs. It is therefore of great interest to know whether this behaviour is accurately captured when they are solved by numerical methods.

Linear Multistep Methods—V: Solving Implicit Methods

The discussion of absolute stability in previous chapters shows that it can be advantageous to use an implicit LMM—usually when the step size in an explicit method has to be chosen on grounds of stability rather than accuracy. One then has to compute the numerical solution at each step by solving a nonlinear system of algebraic equations.

Runge–Kutta Method—I: Order Conditions

Runge–Kutta (RK) methods are one-step methods composed of a number of stages. A weighted average of the slopes (f) of the solution computed at nearby points is used to determine the solution at t = t n+1 from that at t = t n . Euler’s method is the simplest such method and involves just one stage.

Euler’s Method

During the course of this book we will describe three families of methods for numerically solving IVPs: the Taylor series (TS) method, linear multistep methods (LMMs) and Runge–Kutta (RK) methods.

The Taylor Series Method

An alternative is to use a more sophisticated recurrence relation at each step in order to achieve greater accuracy (for the same value of h) or a similar level of accuracy with a larger value of h (and, therefore, fewer steps).

Geometric Integration Part I—Invariants

We judge a numerical method by its ability to “approximate” the ODE. It is perfectly natural to

Linear Multistep Methods—III: Absolute Stability

Thus, convergent methods generate numerical solutions that are arbitrarily close to the exact solution of the IVP provided that h is taken to be sufficiently small. Since non-convergent methods are of little practical use we shall henceforth assume that all LMMs used are convergent—they are consistent and zero-stable.

Geometric Integration Part II—Hamiltonian Dynamics

This chapter continues our study of geometric features of ODEs. We look at Hamiltonian problems, which possess the important property of symplecticness. As in the previous chapter our emphasis is on

Runge-Kutta Methods–II Absolute Stability

The notion of absolute stability developed in Chapter 6 for LMMs is equally relevant to RK methods. Applying an RK method to the linear ODE x′(t) = λx(t) with ℜ(λ) < 0, absolute stability requires that x n → 0 as n → ∞.

Linear Multistep Methods—I: Construction and Consistency

The effectiveness of the family of TS(p) methods has been evident in the preceding chapter. For order p > 1, however, they suffer a serious disadvantage in that they require the right-hand side of the differential equation to be differentiated a number of times. This often rules out their use in real-world applications, which generally involve (large) systems of ODEs whose differentiation is impractical unless automated tools are used [23]. We look, therefore, for alternatives that do not require the use of second and higher derivatives of the solution.