Adaptive time-stepping for incompressible flow part I: scalar advection-diffusion

Even the simplest advection-diffusion problems can exhibit multiple time scales. This means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally. The performance of the second order trapezoid rule using an explicit Adams–Bashforth method for error control is assessed in this work. This combination is particularly well suited to long time integration of advection-dominated problems. Herein it is shown that a stabilized implementation of the trapezoid rule leads to a very effective integrator in other situations: specifically diffusion problems with rough initial data; and general advection-diffusion problems with different physical time scales governing the system evolution

Adaptive time-stepping for incompressible flow. Part II: Navier-Stokes equations

We outline a new class of robust and efficient methods for solving the Navier- Stokes equations. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the potential of our approach. © 2010 Society for Industrial and Applied Mathematics

The reliability of local error estimators for convection-diffusion equations

We assess the reliability of a simple a posteriori error estimator for steady state convection-diffusion equations in cases where convection dominates. Our estimator is computed by solving a local Poisson problem with Neumann boundary conditions. It gives global upper and local lower bounds on the error measured in the $H^1$ semi-norm, except that the error may be over-estimated locally within boundary layers if these are not resolved by the mesh, that is, when the local mesh Péclet number is significantly greater than unity. We discuss the implications of this over-estimation in a practical context where the estimator is used as a local error indicator within a self-adaptive mesh refinement process.\ud \ud This work was supported by EPSRC grants GR/K91262 and GR/L05617

Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity

This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic solid is almost incompressible. Several novel a posteriori error estimators for the energy norm of the finite element error are proposed and analysed. We establish upper and lower bounds for the energy error in terms of the proposed error estimators and prove that the constants in the bounds are independent of the Lam\'{e} coefficients: thus the proposed estimators are robust in the incompressible limit. Numerical results are presented that validate the theoretical estimates. The software used to generate these results is available online.Comment: 23 pages, 9 figure

Radiochemical studies of nuclear fission

The relative yields of 19 nuclides* have been measured in the 14-MeV neutron-induced fission of natural uranium, and have been shown to fall on the familiar type of double- peaked mass-yield curve. The measurements on the two xenon isotopes ((^133)Xe and (^135)Xe) indicate the presence of fine structure in the region of the heavy peak. The mean peak-to-trough ratio is 9.1, which is of the order expected for fission at this energy, and the best fit for the mirror-points is obtained when v, the number of prompt neutrons emitted per fission, is taken to be 4. The condition that the sum of the yields of all the fission-products must be 200% enables values for the absolute yields to be determined: the value so obtained for (^99)Mo is (6.31 ± 0.23)%. A Cockcroft-Walton accelerator was used to produce the 14-MeV neutrons by the D+T reaction. At this neutron energy the cross-sections for (^235)U and (^238)U are of the same order of magnitude, so the results are essentially those for the fission of (^238)U

A preconditioner for the 3D Oseen equations

We describe a preconditioner for the linearised incompressible Navier-Stokes equations (the Oseen equations) which requires as components only a preconditioner/solver for each of a discrete Laplacian and a discrete advection-diffusion operator. With this preconditioner, convergence of an iterative method such as GMRES is independent of the mesh size and depends only mildly on the viscosity parameter (the inverse Reynolds number). Thus when the component preconditioner/solvers are effective on their respective subproblems (as one expects with an appropriate multigrid cycle for instance) a fast Oseen solver results

Australian Sheep Industry CRC: Economic Evaluations of Scientific Research Programs

By the end of its seven-year term in 2007-08, the Australian Sheep Industry CRC (Sheep CRC) will have received total funds of about $90 million, that comprises Commonwealth and industry funding of$30 million, and in-kind contributions valued at $60 million. This level of public and private funding emphasises the need for the Sheep CRC to demonstrate that its research programs will generate sound economic returns to all stakeholders. This paper reports an evaluation of the potential economic value of the achievements of the Sheep CRC at the midpoint of its term of operations at which it has some completed research and a large volume of research in progress. The main question that has been addressed in this evaluation concerns the nature and likely magnitude of the potential benefits relative to the costs of their realisation. The economic methods and other procedures that were used to answer this question, the evaluation scenarios and the results obtained are described. Based on the defined with- and without-Sheep CRC evaluation scenarios, the ‘bottom-line’ result was that the Sheep CRC’s scientific research programs have the potential to deliver a total incremental benefit with a 20-year net present value (NPV) of$191.3 million, and a total benefit-cost ratio (BCR) of 8.1:1 (both at a 5% real rate of discount), indicating that the Sheep CRC’s total research investment over all programs has the potential to return about $8 for every$1 of research investment funds.sheep research, economic evaluations, economic-surplus- benefit-cost analysis., Agribusiness, Farm Management, Livestock Production/Industries, Production Economics, Research and Development/Tech Change/Emerging Technologies, Q160,

A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data

We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of the adaptive multilevel algorithm and we provide optimal choices for error tolerances at each level. Two numerical examples demonstrate the reliability of the error control and the significant decrease in the complexity that arises when compared to single level algorithms and multilevel algorithms that employ adaptivity solely in the spatial discretisation or in the collocation procedure.Comment: 26 pages, 7 figure