Acta Numerica - Current Issue

Volume 16

Mon, 30 Apr 2007 23:00:00 GMT

Acta Numerica, Volume 16

This annual collection of review articles includes survey papers by leading researchers in numerical analysis and scientific computing. The papers present overviews of recent advances and provide state-of-the-art techniques and analysis. Covering the breadth of numerical analysis, articles are written in a style accessible to researchers at all levels and can serve as advanced teaching aids. Broad subject areas for inclusion are computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis and nonlinear dynamical systems, as well as the application of computational techniques in science and engineering and the mathematical theory underlying numerical methods.

Molecular dynamics and the accuracy of numerically computed averages

Research Articles
Stephen D. Bond, Benedict J. Leimkuhler,
Acta Numerica, Volume 16 , pp 1-65

Abstract
Molecular dynamics is discussed from a mathematical perspective. The recent history of method development is briefly surveyed with an emphasis on the use of geometric integration as a guiding principle. The recovery of statistical mechanical averages from molecular dynamics is then introduced, and the use of backward error analysis as a technique for analysing the accuracy of numerical averages is described. This article gives the first rigorous estimates for the error in statistical averages computed from molecular dynamics simulation based on backward error analysis. It is shown that molecular dynamics introduces an appreciable bias at stepsizes which are below the stability threshold. Simulations performed in such a regime can be corrected by use of a stepsize-dependent reweighting factor. Numerical experiments illustrate the efficacy of this approach. In the final section, several open problems in dynamics-based molecular sampling are considered.

Finite volume methods for hyperbolic conservation laws

Research Articles
K. W. Morton, T. Sonar,
Acta Numerica, Volume 16 , pp 155-238

Abstract
Finite volume methods apply directly to the conservation law form of a differential equation system; and they commonly yield cell average approximations to the unknowns rather than point values. The discrete equations that they generate on a regular mesh look rather like finite difference equations; but they are really much closer to finite element methods, sharing with them a natural formulation on unstructured meshes. The typical projection onto a piecewise constant trial space leads naturally into the theory of optimal recovery to achieve higher than first-order accuracy. They have dominated aerodynamics computation for over forty years, but they have never before been the subject of an ActaNumerica article. We shall therefore survey their early formulations before describing powerful developments in both their theory and practice that have taken place in the last few years.

Semi-analytic geometry with R-functions

Research Articles
Vadim Shapiro,
Acta Numerica, Volume 16 , pp 239-303

Abstract
V. L. Rvachev called R-functions because they encode complete logical information within the standard setting of real analysis. He invented them in the 1960s as a means for unifying logic, geometry, and analysis within a common computational framework in an effort to develop a new computationally effective language for modelling and solving boundary value problems. Over the last forty years, R-functions have been accepted as a valuable tool in computer graphics, geometric modelling, computational physics, and in many areas of engineering design, analysis, and optimization. Yet, many elements of the theory of R-functions continue to be rediscovered in different application areas and special situations. The purpose of this survey is to expose the key ideas and concepts behind the theory of R-functions, explain the utility of R-functions in a broad range of applications, and to discuss selected algorithmic issues arising in connection with their use.

Filters, mollifiers and the computation of the Gibbs phenomenon

Research Articles
Eitan Tadmor,
Acta Numerica, Volume 16 , pp 305-378

Abstract
We are concerned here with processing discontinuous functions from their spectral information. We focus on two main aspects of processing such piecewise smooth data: detecting the edges of a piecewise smooth f, namely, the location and amplitudes of its discontinuities; and recovering with high accuracy the underlying function in between those edges. If f is a smooth function, say analytic, then classical Fourier projections recover f with exponential accuracy. However, if f contains one or more discontinuities, its global Fourier projections produce spurious Gibbs oscillations which spread throughout the smooth regions, enforcing local loss of resolution and global loss of accuracy. Our aim in the computation of the Gibbs phenomenon is to detect edges and to reconstruct piecewise smooth functions, while regaining the high accuracy encoded in the spectral data.

Numerical aspects of special functions

Research Articles
Nico M. Temme,
Acta Numerica, Volume 16 , pp 379-478

Abstract
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

Modelling atmospheric flows

Research Articles
Mike Cullen,
Acta Numerica, Volume 16 , pp 67-154

Abstract
This article demonstrates how numerical methods for atmospheric models can be validated by showing that they give the theoretically predicted rate of convergence to relevant asymptotic limit solutions. This procedure is necessary because the exact solution of the Navier Stokes equations cannot be resolved by production models. The limit solutions chosen are those most important for weather and climate prediction. While the best numerical algorithms for this purpose largely reflect current practice, some important limit solutions cannot be captured by existing methods. The use of Lagrangian rather than Eulerian averaging may be required in these cases.