Upwind solution of singular differential equations arising from steady channel flows

We study ordinary nonlinear singular differential equations which arise from steady conservation laws with source terms. An example of steady conservation laws which leads to those scalar equations is the Saint–Venant equations. The numerical solution of these scalar equations is sought by using the ideas of upwinding and discretisation of source terms. Both the Engquist–Osher scheme and the Roe scheme are used with different strategies for discretising the source terms.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.

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.

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.

Eicosapentaenoic acid and docosahexaenoic acid from fish oils: differential associations with lipid responses

Fish-oil supplementation can reduce circulating triacylglycerol (TG) levels and cardiovascular risk. This study aimed to assess independent associations between changes in platelet eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA) and fasting and postprandial (PP) lipoprotein concentrations and LDL oxidation status, following fish-oil intervention. Fiftyfive mildly hypertriacylglycerolaemic (TG 1·5–4·0 mmol/l) men completed a double-blind placebo controlled cross over study, where individuals consumed 6 g fish oil (3 g EPA � DHA) or 6 g olive oil (placebo)/d for two 6-week intervention periods, with a 12-week wash-out period in between. Fish-oil intervention resulted in a significant increase in the platelet phospholipid EPA (+491 %, P,0·001) and DHA (+44 %, P,0·001) content and a significant decrease in the arachidonic acid (210 %, P,0·001) and g-linolenic acid (224 %, P,0·001) levels. A 30% increase in ex vivo LDL oxidation (P,0·001) was observed. In addition, fish oil resulted in a significant decrease in fasting and PP TG levels (P,0·001), PP non-esterified fatty acid (NEFA) levels, and in the percentage LDL as LDL-3 (P�0·040), and an increase in LDLcholesterol (P�0·027). In multivariate analysis, changes in platelet phospholipid DHA emerged as being independently associated with the rise in LDL-cholesterol, accounting for 16% of the variability in this outcome measure (P�0·030). In contrast, increases in platelet EPA were independently associated with the reductions in fasting (P�0·046) and PP TG (P�0·023), and PP NEFA (P�0·015), explaining 15–20% and 25% of the variability in response respectively. Increases in platelet EPA � DHA were independently and positively associated with the increase in LDL oxidation (P�0·011). EPA and DHA may have differential effects on plasma lipids in mildly hypertriacylglycerolaemic men.

Velocity-based moving mesh methods for nonlinear partial differential equations

This article describes a number of velocity-based moving mesh numerical methods formultidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Direct Computation of Stochastic Flow in Reservoirs with Uncertain Parameters

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point and to the field covariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data.