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We report key learning from the public health management of the first two confirmed cases of COVID-19 identified in the UK. The first case imported, and the second associated with probable person-to-person transmission within the UK. Contact tracing was complex and fast-moving. Potential exposures for both cases were reviewed, and 52 contacts were identified. No further confirmed COVID-19 cases have been linked epidemiologically to these two cases. As steps are made to enhance contact tracing across the UK, the lessons learned from earlier contact tracing during the country's containment phase are particularly important and timely.
Finite volume methods apply directly to the conservation law form of a differential equation system; and they commonly yield cell average approximations to the unknowns rather than point values. The discrete equations that they generate on a regular mesh look rather like finite difference equations; but they are really much closer to finite element methods, sharing with them a natural formulation on unstructured meshes. The typical projection onto a piecewise constant trial space leads naturally into the theory of optimal recovery to achieve higher than first-order accuracy. They have dominated aerodynamics computation for over forty years, but they have never before been the subject of an Acta Numerica article. We shall therefore survey their early formulations before describing powerful developments in both their theory and practice that have taken place in the last few years.
This is the 2005 second edition of a highly successful and well-respected textbook on the numerical techniques used to solve partial differential equations arising from mathematical models in science, engineering and other fields. The authors maintain an emphasis on finite difference methods for simple but representative examples of parabolic, hyperbolic and elliptic equations from the first edition. However this is augmented by new sections on finite volume methods, modified equation analysis, symplectic integration schemes, convection-diffusion problems, multigrid, and conjugate gradient methods; and several sections, including that on the energy method of analysis, have been extensively rewritten to reflect modern developments. Already an excellent choice for students and teachers in mathematics, engineering and computer science departments, the revised text includes more latest theoretical and industrial developments.
In the ten years since the first edition of this book was published, the numerical solution of PDEs has moved forward in many ways. But when we sought views on the main changes that should be made for this second edition, the general response was that we should not change the main thrust of the book or make very substantial changes. We therefore aimed to limit ourselves to adding no more than 10%–20% of new material and removing rather little of the original text: in the event, the book has increased by some 23%.
Finite difference methods remain the starting point for introducing most people to the solution of PDEs, both theoretically and as a tool for solving practical problems. So they still form the core of the book. But of course finite element methods dominate the elliptic equation scene, and finite volume methods are the preferred approach to the approximation of many hyperbolic problems. Moreover, the latter formulation also forms a valuable bridge between the two main methodologies. Thus we have introduced a new section on this topic in Chapter 4; and this has also enabled us to reinterpret standard difference schemes such as the Lax–Wendroff method and the box scheme in this way, and hence for example show how they are simply extended to nonuniform meshes.
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.