Definition of the problems considered

In this chapter we shall gather together and formalise definitions that we have introduced in earlier chapters. This will enable us to state and prove the main part of the key Lax Equivalence Theorem. For simplicity we will not aim at full generality but our definitions and arguments will be consistent with those used in a more general treatment.

In the problems which we shall consider, we make the following assumptions:

• the region Ω is a fixed bounded open region in a space which may have one, two, three or more dimensions, with co-ordinates which may be Cartesian (x, y, …), cylindrical polar, spherical polar, etc.;

• the region Ω has boundary ∂Ω;

• the required solution is a function u of the space variables, and of t, defined on Ω × [0, tF]; this function may be vector-valued, so that our discussion can be applied to systems of differential equations, as well as to single equations;

• the operator L(·) involves the partial derivatives of u in the space variables; L does not involve t explicitly; for the most part we shall assume that L is a linear operator, but whenever possible we shall give definitions and state results which will generalise as easily as possible to nonlinear operators;

• the boundary conditions will prescribe the values of g(u) on some or all of the boundary Ω, where g(·) is an operator which may involve spatial partial derivatives;

• the initial condition prescribes the value of u for t = 0 over the region Ω.

The origin of this book was a sixteen-lecture course that each of us has given over the last several years to final-year Oxford undergraduate mathematicians; and its development owes much to the suggestions of some of our colleagues that the subject matter could readily be taught somewhat earlier as a companion course to one introducing the theory of partial differential equations. On the other hand, we have used much of the same material in teaching a one-year Master's course on mathematical modelling and numerical analysis. These two influences have guided our choice of topics and the level and manner of presentation.

Thus we concentrate on finite difference methods and their application to standard model problems. This allows the methods to be couched in simple terms while at the same time treating such concepts as stability and convergence with a reasonable degree of mathematical rigour. In a more advanced text, or one with greater emphasis on the finite element method, it would have been natural and convenient to use standard Sobolev space norms. We have avoided this temptation and used only discrete norms, specifically the maximum and the l2 norms. There are several reasons for this decision. Firstly, of course, it is consistent with an aim of demanding the minimum in prerequisites – of analysis, of PDE theory, or of computing – so allowing the book to be used as a text in an early undergraduate course and for teaching scientists and engineers as well as mathematicians.

Over the past several decades, molecular simulation has become increasingly important for chemists, physicists, bio-scientists, and engineers, and plays a role in applications such as rational drug design and the development of new types of materials. While many levels of detail can be incorporated if desired, in most cases work is performed with a simplified atomic model, consisting of a large number of mass points interacting in various types of forces, i.e. an N-body problem.

There are essentially two principal types of simulation methodology in common use. In Monte-Carlo (MC) methods, random steps are taken in order to achieve a rapid sampling of the most likely states of the molecule. In molecular dynamics (MD), the idea is to construct approximate trajectories for the N-body problem and to use these to gain an understanding of how the molecule evolves in time, for example in response to a stimulus, during a transition between states, or as a means for calculating averages. It should be stressed that only MD and not MC methods allow the theoretical possibility of obtaining time-dependent quantities from simulation, while both schemes can in principle be used for the same statistical-mechanical calculations. Increasingly, one finds that MD and MC schemes are combined in various ways to seek improved efficiency. In this chapter we will focus only on (pure) MD methods, and in particular on the geometric integration issues associated to computing MD trajectories. For a more complete perspective on molecular simulation, the reader is referred to a text on the subject such as that of Schlick, Allen and Tildesley, Rappaport, or Frenkel and Smit.

Hamiltonian systems often exhibit dynamical phenomena covering a vast range of different time scales. In this chapter, we will discuss systems with two well separated time scales. More specifically, we consider systems for which the fast motion is essentially oscillatory. Such systems can arise from very different applications such as celestial or molecular dynamics and they might manifest themselves in very different types of Hamiltonian equations. Hence, the discussion in this chapter is necessarily limited to special cases. However, the basic principles and ideas have a much wider range of applicability.

A standard integrator, whether symplectic or not, will, in general, have to use a stepsize that resolves the oscillations in the fast system and, hence, one might be forced to use very small timesteps in comparison to the slow dynamics which is of primary interest. However, in special cases, one might be able to individually exactly solve the fast oscillatory and the slow system. Following the idea of splitting methods, this suggests to compose these two exact propagators and to apply a stepsize that is large with respect to the period of the fast oscillations. Such a method is then called a large timestep (LTS) method. Often the fast oscillations cannot be integrated analytically. A natural idea for the construction of an LTS method is then to assign different timesteps to different parts of the system. This approach is called multiple timestepping (MTS) and can often even be implemented such that the overall timestepping procedure still generates a symplectic map. We will explain the basic idea of symplectic LTS/MTS methods in Section 10.1.