An algorithm for the shallow water equations with body fitted meshes

An efficient algorithm based on flux difference splitting is presented for the solution of the two-dimensional shallow water equations in a generalised coordinate system. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The scheme has good jump capturing properties and the advantage of using body-fitted meshes. Numerical results are shown for flow past a circular obstruction.

Prediction of supercritical flow in open channels

A finite difference scheme based on flux difference splitting is presented for the solution of the one-dimensional shallow water equations in open channels. A linearised problem, analogous to that of Riemann for gas dynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearised problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids non-physical, spurious oscillations. The scheme is applied to a problem of flow in a river whose geometry induces a region of supercritical flow.

Electronic voting systems in undergraduate teaching

One of the major differences undergraduates experience during the transition to university is the style of teaching. In schools and colleges most students study key stage 5 subjects in relatively small informal groups where teacher–pupil interaction is encouraged and two-way feedback occurs through question and answer type delivery. On starting in HE students are amazed by the sizes of the classes. For even a relatively small chemistry department with an intake of 60-70 students, biologists, pharmacists, and other first year undergraduates requiring chemistry can boost numbers in the lecture hall to around 200 or higher. In many universities class sizes of 400 are not unusual for first year groups where efficiency is crucial. Clearly the personalised classroom-style delivery is not practical and it is a brave student who shows his ignorance by venturing to ask a question in front of such an audience. In these environments learning can be a very passive process, the lecture acts as a vehicle for the conveyance of information and our students are expected to reinforce their understanding by ‘self-study’, a term, the meaning of which, many struggle to understand. The use of electronic voting systems (EVS) in such situations can vastly change the students’ learning experience from a passive to a highly interactive process. This principle has already been demonstrated in Physics, most notably in the work of Bates and colleagues at Edinburgh.1 These small hand-held devices, similar to those which have become familiar through programmes such as ‘Who Wants to be a Millionaire’ can be used to provide instant feedback to students and teachers alike. Advances in technology now allow them to be used in a range of more sophisticated settings and comprehensive guides on use have been developed for even the most techno-phobic staff.

Optimal control of nonlinear systems represented by differential algebraic equations

An iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified simplified model based problem with parameter updating in such a manner that the correct solution of the original nonlinear problem is achieved.

Generalised Dirichelt-to-Neumann map in time dependent domains

We study the heat, linear Schrodinger and linear KdV equations in the domain l(t) < x < ∞, 0 < t < T, with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.

Boundary value problems for the elliptic sine-Gordon equation in a semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2 by 2 matrix Riemann-Hilbert problem whose \jump matrix" depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however a major difficulty for this problem is the existence of non-integrable singularities of the function q_y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in terms of a 2 by 2 matrix Riemann-Hilbert problem whose jump matrix depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h. The determination of the function h remains open.

On the time-dependent analysis of Gamow decay

Gamow's explanation of the exponential decay law uses complex 'eigenvalues' and exponentially growing 'eigenfunctions'. This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wavefunctions. Observing that the time evolution of any wavefunction is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover, the presentation can well be used in physics lectures to students.

Correlations between two sets of angular relation equations

It is shown here that the angular relation equations between direct and reciprocal vectors are very similar to the angular relation equations in Euler's theorem. These two sets of equations are usually treated separately as unrelated equations in different fields. In this careful study, the connection between the two sets of angular equations is revealed by considering the cosine rule for the spherical triangle. It is found that understanding of the correlation is hindered by the facts that the same variables are defined differently and different symbols are used to represent them in the two fields. Understanding the connection between different concepts is not only stimulating and beneficial, but also a fundamental tool in innovation and research, and has historical significance. The background of the work presented here contains elements of many scientific disciplines. This work illustrates the common ground of two theories usually considered separately and is therefore of benefit not only for its own sake but also to illustrate a general principle that a theory relevant to one discipline can often be used in another. The paper works with chemistry related concepts using mathematical methodologies unfamiliar to the usual audience of mainstream experimental and theoretical chemists.

Coupled Kelvin-wave and mirage-wave instabilities in semigeostrophic dynamics

A weak instability mode, associated with phase-locked counterpropagating coastal Kelvin waves in horizontal anticyclonic shear, is found in the semigeostrophic (SG) equations for stratified flow in a channel. This SG instability mode approximates a similar mode found in the Euler equations in the limit in which particle-trajectory slopes are much smaller than f/N, where f is the Coriolis frequency and N > f the buoyancy frequency. Though weak under normal parameter conditions, this instability mode is of theoretical interest because its existence accounts for the failure of an Arnol’d-type stability theorem for the SG equations. In the opposite limit, in which the particle motion is purely vertical, the Euler equations allow only buoyancy oscillations with no horizontal coupling. The SG equations, on the other hand, allow a physically spurious coastal “mirage wave,” so called because its velocity field vanishes despite a nonvanishing disturbance pressure field. Counterpropagating pairs of these waves can phase-lock to form a spurious “mirage-wave instability.” Closer examination shows that the mirage wave arises from failure of the SG approximations to be self-consistent for trajectory slopes f/N.

The physical pendulum in an advanced undergraduate course in mechanics

The physical pendulum treated with a Hamiltonian formulation is a natural topic for study in a course in advanced classical mechanics. For the past three years, we have been offering a series of problem sets studying this system numerically in our third-year undergraduate courses in mechanics. The problem sets investigate the physics of the pendulum in ways not easily accessible without computer technology and explore various algorithms for solving mechanics problems. Our computational physics is based on Mathematica with some C communicating with Mathematica, although nothing in this paper is dependent on that choice. We have nonetheless found this system, and particularly its graphics, to be a good one for use with undergraduates.