The technique of obtaining high resolution, second order, oscillation free (TVD), explicit scalar difference schemes, by the addition of a limited antidiffusive flux to a first order scheme is explored and bounds derived for such limiters. A class of limiters is presented which includes a very compressive limiter due to Roe, and various limiters are compared both theoretically and numerically.

SUMMARYThis paper investigates how to accurately numerically approximate the equations governing 2D sediment transport by considering two approaches: a steady and unsteady approach. A high-resolution scheme based on Roe's scheme is used to approximate both approaches with the results compared for a 2D test case.

Cholesterol is one of the key constituents for maintaining the cellular membrane and thus the integrity of the cell itself. In contrast high levels of cholesterol in the blood are known to be a major risk factor in the development of cardiovascular disease. We formulate a deterministic nonlinear ordinary differential equation model of the sterol regulatory element binding protein 2 (SREBP-2) cholesterol genetic regulatory pathway in a hepatocyte. The mathematical model includes a description of genetic transcription by SREBP-2 which is subsequently translated to mRNA leading to the formation of 3-hydroxy-3-methylglutaryl coenzyme A reductase (HMGCR), a main regulator of cholesterol synthesis. Cholesterol synthesis subsequently leads to the regulation of SREBP-2 via a negative feedback formulation. Parameterised with data from the literature, the model is used to understand how SREBP-2 transcription and regulation affects cellular cholesterol concentration. Model stability analysis shows that the only positive steady-state of the system exhibits purely oscillatory, damped oscillatory or monotic behaviour under certain parameter conditions. In light of our findings we postulate how cholesterol homeostasis is maintained within the cell and the advantages of our model formulation are discussed with respect to other models of genetic regulation within the literature.

The goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms. This interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as "the dynamics of numerics and the numerics of dynamics." At the present time, this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications. This is the first of a series of research papers under the same topic. Our hope is to reach researchers in the fields of computational fluid dynamics (CFD) and, in particular, hypersonic and combustion related CFD. By simple examples (in which the exact solutions of the governing equations are known), the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results. One striking phenomenon is that with the same initial data, the continuum and its discretized counterpart can asymptotically approach different stable solutions. This behavior is especially important for employing a time-dependent approach to the steady state since the initial data are usually not known and a freest ream condition or an intelligent guess for the initial conditions is often used. With the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart, it is not easy to delineate the * An abbreviated version appeared in the "All rights of reproduction in any form reserved. 250YEE, SWEBY, AND GRIFFITHS true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum. Part I concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in CFD and to serve as an introduction for parts II and III of the same series of research papers. .: [)

The bifurcation diagram associated with the logistic equation v" +1 = av"(\-v") is by now well known, as is its equivalence to solving the ordinary differential equation (ODE) u' = crw(l-u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. We investigate, both analytically and computationally, Runge-Kutta schemes applied to the equation u'=f\u), for f(u) =