Time-decoupled high order continuous space-time finite element schemes for the heat equation

Copyright © by SIAMIn Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 6685—6708 Werder et al. demonstrated that time discretizations of the heat equation by a temporally discontinuous Galerkin finite element method could be decoupled by diagonalising the temporal ‘Gram matrices’. In this article we propose a companion approach for the heat equation by using a continuous Galerkin time discretization. As a result, if piecewise polynomials of degree d are used as the trial functions in time and the spatial discretization produces systems of dimension M then, after decoupling, d systems of size M need to be solved rather than a single system of sizeMd. These decoupled systems require complex arithmetic, as did Werder et al.’s technique, but are amenable to parallel solution on modern multi-core architectures. We give numerical tests for temporal polynomial degrees up to six for three different model test problems, using both Galerkin and spectral element spatial discretizations, and show convergence and temporal superconvergence rates that accord with the bounds given by Aziz and Monk, Math. Comp. 52:186 (1989), pp. 255—274. We also interpret error as a function of computational time and see that our high order schemes may offer greater efficiency that the Crank-Nicolson method in terms of accuracy per unit of computational time—although in a multi-core world, with highly tuned iterative solvers, one has to be cautious with such claims. We close with a speculation on the application of these ideas to the Navier-Stokes equations for incompressible fluids

Modification of cosmic-ray energy spectra by stochastic acceleration

Context: Typical space plasmas contain spatially and temporally variable turbulent electromagnetic fields. Understanding the transport of energetic particles and the acceleration mechanisms for charged particles is an important goal of today's astroparticle physics. Aims: To understand the acceleration mechanisms at the particle source, subsequent effects have to be known. Therefore, the modification of a particle energy distribution, due to stochastic acceleration, needs to be investigated. Methods: The diffusion in momentum space was investigated by using both a Monte-Carlo simulation code and by analytically solving the momentum-diffusion equation. For simplicity, the turbulence was assumed to consist of one-dimensional Alfven waves. Results: Using both methods, it is shown that, on average, all particles with velocities comparable to the Alfven speeds are accelerated. This influences the energy distribution by significantly increasing the energy spectral index. Conclusions: Because of electromagnetic turbulence, a particle energy spectrum measured at Earth can drastically deviate from its initial spectrum. However, for particles with velocities significantly above the Alfven speed, the effect becomes negligible.Comment: 10 pages, 6 figures, accepted for publication in Astronomy & Astrophysic

Computer program to assess impact of fatigue and fracture criteria on weight and cost of transport aircraft

A preliminary design analysis tool for rapidly performing trade-off studies involving fatigue, fracture, static strength, weight, and cost is presented. Analysis subprograms were developed for fatigue life, crack growth life, and residual strength; and linked to a structural synthesis module which in turn was integrated into a computer program. The part definition module of a cost and weight analysis program was expanded to be compatible with the upgraded structural synthesis capability. The resultant vehicle design and evaluation program is named VDEP-2. It is an accurate and useful tool for estimating purposes at the preliminary design stage of airframe development. A sample case along with an explanation of program applications and input preparation is presented

Information Content in Data Sets for a Nucleated-Polymerization Model

We illustrate the use of tools (asymptotic theories of standard error quantification using appropriate statistical models, bootstrapping, model comparison techniques) in addition to sensitivity that may be employed to determine the information content in data sets. We do this in the context of recent models [23] for nucleated polymerization in proteins, about which very little is known regarding the underlying mechanisms; thus the methodology we develop here may be of great help to experimentalists