Ronald Smith

Derivative formulae and errors for non-uniformly spaced points

Lagrange interpolation estimates a function at a reference position Χ from known values of the function at distinct non-uniformly spaced points x1, …, xn. Here, the corresponding n-point finite-difference formulae are derived to estimate derivatives up to order n−1 at Χ. A recurrence relation is derived that permits the errors to be determined as a Taylor series to any accuracy. The error coefficient multiplying the n+jth derivative is a polynomial of order j+1 in the elementary symmetric functions for the displacements x1−Χ,…,xn−Χ. Appendices state finite-difference formulae to estimate the derivatives and the first four error terms for n=1,…,5.

Optimal and near-optimal advection—diffusion finite-difference schemes III. Black—Scholes equation

Optimal and near–optimal compact finite–difference schemes are presented and tested for the numerical solution of an extended Black–Scholes equation: ∂ t c-rc+rs ∂ s c+ 1 2 σ 2 s 2 ∂ s 2 c=-q-s ∂ s (s ∂ s g). Here c(s,t) is the expected value of the right to buy or sell an asset at some future date, s is the asset price, r(t) is the rate of increase available from alternative riskless investments, and σ(t) is the asset volatility. The terms on the right–hand side allow the applicability to be extended beyond the basic European options model. The compactness of the numerical scheme keeps any computer programming elementary. The required computational resources can be as small as 0.001 of conventional schemes.

Two outfalls in an estuary : Optimal wasteload allocation

When two outfalls are discharging wastewater into a narrow (rapidly mixed) estuary within a tidal excursion of each other, the pollutant concentrations experienced at the two outfall sites are strongly inter-dependent. It is shown how a given total tidally integrated effluent load can be allocated optimally between the two outfalls so that the peak concentration (in time and position) of the principal contaminant species is minimized. Graphical results show the dependence of the wasteload allocation and of the peak concentration upon the pollutant decay rate, the separation between the outfalls and the fresh water flow along the estuary. Optimization with respect to any one of a mixture of pollutants is close to optimal for a wide range of other pollutants.

Dispersion far downstream of a river junction

The merging of two rivers can have substantial and persistent effects upon the dispersion of a sudden release of pollutant from one of the tributaries. This paper quantifies the delayed time of arrival and increase in spreading attributable to the junction.

Optimal and near–optimal advection–diffusion finite–difference schemes. V. Error propagation

The wave concept of group (or energy) velocity is used to predict how errors propagate in three numerical schemes for the solution of the diffusion equation with flow and decay. A five–point filter is used to isolate the long and the saw–tooth grid–scale errors. Contour plots are presented which illustrate the markedly different propagation of the long and saw–tooth errors. The energy of long errors is carried downstream with the flow. An illustrative example is given for which, as predicted, the saw–tooth errors go upstream.

Shear dispersion along a rotating axle in a closely fitting shaft

A formula is derived for the longitudinal shear dispersion coefficient of a solute in a laminar flow along and around a rotating cylindrical axle in an off-centre closely fitting shaft. The rotation drives a circulation which augments diffusive mixing around the axle and reduces the eventual rate of longitudinal spreading. A simple approximation is shown to give accurate results for the important special case of a cylindrical shaft

Multi-mode models of flow and of solute dispersion in shallow water. Part 3. Horizontal dispersion tensor for velocity

Svendsen & Putrevu (1994) revealed that the much larger off-shore than vertical effective viscosity for longshore currents is the consequence of a shear dispersion mechanism. The multi-mode representation for the flow gives a mathematical framework within which a more general derivation can be made. It is shown that to a first approximation the horizontal shear dispersion tensor for velocity is the same as that for solutes.

Horizontal fractionation of rising and sinking particles in wind-affected currents

The different rise or sinking velocity for different sizes or types of particles gives different vertical sampling of a wind-affected shallow-water flow. This paper derives a mathematical model for the consequent horizontal fractionation of a dilute suspension of particles when the flow is a wind-influenced perturbation from the classical logarithmic open-channel flow. Simple approximations are given for the effective horizontal velocity and for the shear dispersion tensor which preserve the perfect duality between the sensitivity of sinking particles to bed stress and the sensitivity of rising particles to surface stress.

Wind-augmented transport and dilution in shallow-water flows

This paper focuses attention on the joint dependence of the horizontal dilution rates upon the strength of the wind-augmented current and upon the vertical rise (or sinking) velocity of the particles. In strong wind the greatly enhanced mixing counterbalances the onshore drift and explains why shoreline pollution is not significantly correlated to the onshore wind

Two-dimensional shear dispersion for skewed flows in narrow gaps between moving surfaces

Flows transporting material between nearby moving surfaces are ubiquitous in machinery of all scales and with a variety of geometries. Here a general derivation is given of the effective two-dimensional mixing process in a narrow gap for a solute or miscible fluids. Explicit formulae are given for the shear dispersion tensor in laminar and turbulent (logarithmic velocity profile) flows. It is shown that if improved mixing is required, then the optimum direction for additional boundary motion or stress is at right angles to the primary flow direction.