An efficient numerical scheme for the Euler equations

A finite difference scheme based on flux difference splitting is presented for the solution of the Euler equations for the compressible flow of an ideal gas. A linearised Riemann problem is defined, and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency, leading to arithmetic averaging. This is in contrast to the usual ‘square root’ averages found in this type of Riemann solver, where the computational expense can be prohibitive. 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 nonphysical, spurious oscillations. The scheme is applied to a shock tube problem and a blast wave problem. Each approximate solution compares well with those given by other schemes, and for the shock tube problem is in agreement with the exact solution.

An efficient finite difference scheme for three-dimensional, non-equilibrium flows using operator splitting

An efficient finite difference scheme is presented for the inviscid terms of the three-dimensional, compressible flow equations for chemical non-equilibrium gases. This scheme represents an extension and an improvement of one proposed by the author, and includes operator splitting.

A TVD finite difference scheme with non-uniform meshes and without upstream weighting

We present a finite difference scheme, with the TVD (total variation diminishing) property, for scalar conservation laws. The scheme applies to non-uniform meshes, allowing for variable mesh spacing, and is without upstream weighting. When applied to systems of conservation laws, no scalar decomposition is required, nor are any artificial tuning parameters, and this leads to an efficient, robust algorithm.

A finite difference scheme for inviscid flows with non-eequilibrium chemistry and internal energy

A finite difference scheme is presented for the inviscid terms of the equations of compressible fluid dynamics with general non-equilibrium chemistry and internal energy.

A finite difference scheme for steady, supersonic, two-dimensional, compressible flow of real gases

A finite difference scheme is presented for the solution of the two-dimensional equations of steady, supersonic, compressible flow of real gases. The scheme incorparates numerical characteristic decomposition, is shock-capturing by design and incorporates space-marching as a result of the assumption that the flow is wholly supersonic in at least one space dimension. Results are shown for problems involving oblique hydraulic jumps and reflection from a wall.

Flux difference splitting for open channel flows

A finite difference scheme based on flux difference splitting is presented for the solution of the one-dimensional shallow-water equations in open channels, together with an extension to two-dimensional flows. A linearized 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 linearized 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 one-dimensional dam-break problem, and to a problem of flow in a river whose geometry induces a region of supercritical flow. The scheme is also applied to a two-dimensional dam-break problem. The numerical results are compared with the exact solution, or other numerical results, where available.

Ocean acoustic models for low frequency propagation in 2D and 3D environments

In this paper we are mainly concerned with the development of efficient computer models capable of accurately predicting the propagation of low-to-middle frequency sound in the sea, in axially symmetric (2D) and in fully 3D environments. The major physical features of the problem, i.e. a variable bottom topography, elastic properties of the subbottom structure, volume attenuation and other range inhomogeneities are efficiently treated. The computer models presented are based on normal mode solutions of the Helmholtz equation on the one hand, and on various types of numerical schemes for parabolic approximations of the Helmholtz equation on the other. A new coupled mode code is introduced to model sound propagation in range-dependent ocean environments with variable bottom topography, where the effects of an elastic bottom, of volume attenuation, surface and bottom roughness are taken into account. New computer models based on finite difference and finite element techniques for the numerical solution of parabolic approximations are also presented. They include an efficient modeling of the bottom influence via impedance boundary conditions, they cover wide angle propagation, elastic bottom effects, variable bottom topography and reverberation effects. All the models are validated on several benchmark problems and versus experimental data. Results thus obtained were compared with analogous results from standard codes in the literature.

Evaluation of agri-environmental and forestry schemes with multiple objectives

The evaluation of EU policy in the area of rural land use management often encounters problems of multiple and poorly articulated objectives. Agri-environmental policy has a range of aims, including natural resource protection, biodiversity conservation and the protection and enhancement of landscape quality. Forestry policy, in addition to production and environmental objectives, increasingly has social aims, including enhancement of human health and wellbeing, lifelong learning, and the cultural and amenity value of the landscape. Many of these aims are intangible, making them hard to define and quantify. This article describes two approaches for dealing with such situations, both of which rely on substantial participation by stakeholders. The first is the Agri-Environment Footprint Index, a form of multi-criteria participatory approach. The other, applied here to forestry, has been the development of ‘multi-purpose’ approaches to evaluation, which respond to the diverse needs of stakeholders through the use of mixed methods and a broad suite of indicators, selected through a participatory process. Each makes use of case studies and involves stakeholders in the evaluation process, thereby enhancing their commitment to the programmes and increasing their sustainability. Both also demonstrate more ‘holistic’ approaches to evaluation than the formal methods prescribed in the EU Common Monitoring and Evaluation Framework.

Stability of solutions of linear difference equations with periodic coefficients

This paper represents the last technical contribution of Professor Patrick Parks before his untimely death in February 1995. The remaining authors of the paper, which was subsequently completed, wish to dedicate the article to Patrick. A frequency criterion for the stability of solutions of linear difference equations with periodic coefficients is established. The stability criterion is based on a consideration of the behaviour of a frequency hodograph with respect to the origin of coordinates in the complex plane. The formulation of this criterion does not depend on the order of the difference equation.

Selection of funding schemes by a borrowing decision model: a Hong Kong case study

In financial decision-making, a number of mathematical models have been developed for financial management in construction. However, optimizing both qualitative and quantitative factors and the semi-structured nature of construction finance optimization problems are key challenges in solving construction finance decisions. The selection of funding schemes by a modified construction loan acquisition model is solved by an adaptive genetic algorithm (AGA) approach. The basic objectives of the model are to optimize the loan and to minimize the interest payments for all projects. Multiple projects being undertaken by a medium-size construction firm in Hong Kong were used as a real case study to demonstrate the application of the model to the borrowing decision problems. A compromise monthly borrowing schedule was finally achieved. The results indicate that Small and Medium Enterprise (SME) Loan Guarantee Scheme (SGS) was first identified as the source of external financing. Selection of sources of funding can then be made to avoid the possibility of financial problems in the firm by classifying qualitative factors into external, interactive and internal types and taking additional qualitative factors including sovereignty, credit ability and networking into consideration. Thus a more accurate, objective and reliable borrowing decision can be provided for the decision-maker to analyse the financial options.

Exploring the land–ocean contrast in convective vigor using islands

Moist convection is well known to be generally more intense over continental than maritime regions, with larger updraft velocities, graupel, and lightning production. This study explores the transition from maritime to continental convection by comparing the trends in Tropical Rainfall Measuring Mission (TRMM) radar and microwave (37 and 85 GHz) observations over islands of increasing size to those simulated by a cloud-resolving model. The observed storms were essentially maritime over islands of <100 km2 and continental over islands >10 000 km2, with a gradual transition in between. Equivalent radar and microwave quantities were simulated from cloud-resolving runs of the Weather Research and Forecasting model via offline radiation codes. The model configuration was idealized, with islands represented by regions of uniform surface heat flux without orography, using a range of initial sounding conditions without strong horizontal winds or aerosols. Simulated storm strength varied with initial sounding, as expected, but also increased sharply with island size in a manner similar to observations. Stronger simulated storms were associated with higher concentrations of large hydrometeors. Although biases varied with different ice microphysical schemes, the trend was similar for all three schemes tested and was also seen in 2D and 3D model configurations. The successful reproduction of the trend with such idealized forcing supports previous suggestions that mesoscale variation in surface heating—rather than any difference in humidity, aerosol, or other aspects of the atmospheric state—is the main reason that convection is more intense over continents and large islands than over oceans. Some dynamical storm aspects, notably the peak rainfall and minimum surface pressure low, were more sensitive to surface forcing than to the atmospheric sounding or ice scheme. Large hydrometeor concentrations and simulated microwave and radar signatures, however, were at least as sensitive to initial humidity levels as to surface forcing and were more sensitive to the ice scheme. Issues with running the TRMM simulator on 2D simulations are discussed, but they appear to be less serious than sensitivities to model microphysics, which were similar in 2D and 3D. This supports the further use of 2D simulations to economically explore modeling uncertainties.