Convergence analysis of Crank-Nicolson and Rannacher time-marching

This paper presents a convergence analysis of Crank-Nicolson and Rannacher time-marching methods which are often used in finite difference discretisations of the Black-Scholes equations. Particular attention is paid to the important role of Rannacher's startup procedure, in which one or more initial timesteps use Backward Euler timestepping, to achieve second order convergence for approximations of the first and second derivatives. Numerical results confirm the sharpness of the error analysis which is based on asymptotic analysis of the behaviour of the Fourier transform. The relevance to Black-Scholes applications is discussed in detail, with numerical results supporting recommendations on how to maximise the accuracy for a given computational cost

Sharp error estimates for discretisations of the 1D convection/diffusion equation with Dirac initial data

This paper derives sharp estimates of the error arising from explicit and implicit approximations of the constant coefficient 1D convection/diffusion equation with Dirac initial data. The error analysis is based on Fourier analysis and asymptotic approximation of the integrals resulting from the inverse Fourier transform. This research is motivated by applications in computational finance and the desire to prove convergence of approximations to adjoint partial differential equations

Some reports of snowfall from fog during the UK winter of 2008/09

Snowfall during anticyclonic, non-frontal, and foggy conditions is surprising. Because it is often not forecast, it can present a hazard to transport and modify the surface albedo. In this report, we present some observations of snowfall during conditions of freezing fog in the UK during the winter of 2008/09

Parastatistics of charged bosons partly localised by impurities

Coulomb repulsion is taken into account to derive the thermodynamics of charged bosons in a random external potential. A simple analytical form of the partition function is proposed for the case of non-overlapping localised states (i.e. a small amount of disorder). The density of localised bosons and the specific heat show a peculiar non-uniform temperature dependence below the Bose-Einstein condensation temperature. The superfluid - Bose-glass phase diagram is discussed. A new phase is predicted which is a Bose-glass at T=0 but a superfluid at finite T.Comment: 6 pages, Revtex, 6 figures (available on request from the authors

Electromagnetic atmosphere-plasma coupling: the global atmospheric electric circuit

A description is given of the global atmospheric electric circuit operating between the Earth’s surface and the ionosphere. Attention is drawn to the huge range of horizontal and vertical spatial scales, ranging from 10−9 m to 1012 m, concerned with the many important processes at work. A similarly enormous range of time scales is involved from 10−6 s to 109 s, in the physical effects and different phenomena that need to be considered. The current flowing in the global circuit is generated by disturbed weather such as thunderstorms and electrified rain/shower clouds, mostly occurring over the Earth’s land surface. The profile of electrical conductivity up through the atmosphere, determined mainly by galactic cosmic ray ionization, is a crucial parameter of the circuit. Model simulation results on the variation of the ionospheric potential, ∼250 kV positive with respect to the Earth’s potential, following lightning discharges and sprites are summarized. Experimental results comparing global circuit variations with the neutron rate recorded at Climax, Colorado, are then discussed. Within the return (load) part of the circuit in the fair weather regions remote from the generators, charge layers exist on the upper and lower edges of extensive layer clouds; new experimental evidence for these charge layers is also reviewed. Finally, some directions for future research in the subject are suggested

The chaos machine: analogue computing rediscovered (1)

Analogue computers provide actual rather than virtual representations of model systems. They are powerful and engaging computing machines that are cheap and simple to build. This two-part Retronics article helps you build (and understand!) your own analogue computer to simulate the Lorenz butterfly that's become iconic for Chaos theory

Remote sensing of cloud base charge

Layer clouds are abundant in the Earth's atmosphere. Such clouds do not become sufficiently strongly charged to generate lightning, but they show weak charging along the upper and lower cloud boundaries where there is a conductivity transition. Cloud edge charging has recently been observed using balloon-carried electrometers. Measurement of cloud boundary charging without balloons is shown to be possible here for low altitude (<1km) charged cloud bases, through combining their effect on the surface electric field with laser time of flight cloud base height measurements, and the application of simple electrostatic models.Comment: Proceedings of the Electrostatics Society of America conference, Ottawa, June 201