Past Computational Mathematics and Applications Seminar

2 May 2002
14:00
Prof Tim Barth
Abstract
A-Posteriori Error estimates for high order Godunov finite volume methods are presented which exploit the two solution representations inherent in the method, viz. as piecewise constants $u_0$ and cell-wise $q$-th order reconstructed functions $R^0_q u_0$. The analysis provided here applies directly to schemes such as MUSCL, TVD, UNO, ENO, WENO or any other scheme that is a faithful extension of Godunov's method to high order accuracy in a sense that will be made precise. Using standard duality arguments, we construct exact error representation formulas for derived functionals that are tailored to the class of high order Godunov finite volume methods with data reconstruction, $R^0_q u_0$. We then consider computable error estimates that exploit the structure of higher order Godunov finite volume methods. The analysis technique used in this work exploits a certain relationship between higher order Godunov methods and the discontinuous Galerkin method. Issues such as the treatment of nonlinearity and the optional post-processing of numerical dual data are also discussed. Numerical results for linear and nonlinear scalar conservation laws are presented to verify the analysis. Complete details can be found in a paper appearing in the proceedings of FVCA3, Porquerolles, France, June 24-28, 2002.
  • Computational Mathematics and Applications Seminar
Dr Stefano Salvini
Abstract
SMP (Symmetric Multi-Processors) hardware technologies are very popular with vendors and end-users alike for a number of reasons. However, true shared memory parallelism has experienced somewhat slower to take up amongst the scientific-programming community. NAG has been at the forefront of SMP technology for a number of years, and the NAG SMP Library has shown the potential of SMP systems. \\ \\ At the very high end, SMP hardware technologies are used as building blocks of modern supercomputers, which truly consist of clusters of SMP systems, for which no dedicated model of parallelism yet exists. \\ \\ The aim of this talk is to introduce SMP systems and their potential. Results from our work at NAG will also be introduced to show how SMP parallelism, based on a shared memory paradigm, can be used to very good effect and can produce high performance, scalable software. The talk also aims to discuss some aspects of the apparent slow take up of shared memory parallelism and the potential competition from PC (i.e. Intel)-based cluster technology. The talk then aims to explore the potential of SMP technology within "hybrid parallelism", i.e. mixed distributed and shared memory modes, illustrating the point with some preliminary work carried out by the author and others. Finally, a number of potential future challenges to numerical analysts will be discussed. \\ \\ The talk is aimed at all who are interested in SMP technologies for numerical computing, irrespective of any previous experience in the field. The talk aims to stimulate discussion, by presenting some ideas, backing these with data, not to stifle it in an ocean of detail!
  • Computational Mathematics and Applications Seminar
7 March 2002
14:00
Dr Alison Ramage and Prof Howard Elman
Abstract
It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. For the Galerkin finite element method with linear elements on a uniform 1D grid, a precise statement as to exactly when such oscillations occur can be made, namely, that for a problem with mesh size h, constant advective velocity and different values at the left and right boundaries, oscillations will occur if the mesh P\'{e}clet number $P_e$ is greater than one. In 2D, however, the situation is not so well understood. In this talk, we present an analysis of a 2D model problem on a square domain with grid-aligned flow which enables us to clarify precisely when oscillations occur, and what can be done to prevent them. We prove the somewhat surprising result that there are oscillations in the 2D problem even when $P_e$ is less than 1. Also, we show that there are distinct effects arising from differences in the top and bottom boundary conditions (equivalent to those seen in 1D), and the non-zero boundaries parallel to the flow direction.
  • Computational Mathematics and Applications Seminar
Abstract
Algebra based modeling systems are becoming essential elements in the application of large and complex mathematical programs. These systems enable the abstraction, expression and translation of practical problems into reliable and effective operational systems. They provide the bridged between algorithms and real world problems by automating the problem analysis and translation into specific data structures and provide computational services required by different solvers. The modeling system GAMS will be used to illustrate the design goals and main features of such systems. Applications in use and under development will be used to provide the context for discussing the changes in user focus and future requirements. This presents new sets of opportunities and challenges to the supplier and implementer of mathematical programming solvers and modeling systems.
  • Computational Mathematics and Applications Seminar
14 February 2002
14:00
Dr Roland Becker
Abstract
A systematic approach to error control and mesh adaptation for optimal control of systems governed by PDEs is presented. Starting from a coarse mesh, the finite element spaces are successively enriched in order to construct suitable discrete models. This process is guided by an a posteriori error estimator which employs sensitivity factors from the adjoint equation. We consider different examples with the stationary Navier-Stokes equations as state equation.
  • Computational Mathematics and Applications Seminar
Prof Ivan Graham
Abstract
When steady solutions of complex physical problems are computed numerically it is often crucial to compute their <em>stability</em> in order to, for example, check that the computed solution is "physical", or carry out a sensitivity analysis, or help understand complex nonlinear phenomena near a bifurcation point. Usually a stability analysis requires the solution of an <em>eigenvalue problem</em> which may arise in its own right or as an appropriate linearisation. In the case of discretized PDEs the corresponding matrix eigenvalue problem will often be of generalised form: \\ $Ax=\lambda Mx$ (1) \\ with $A$ and $M$ large and sparse. In general $A$ is unsymmetric and $M$ is positive semi-definite. Only a small number of "dangerous" eigenvalues are usually required, often those (possibly complex) eigenvalues nearest the imaginary axis. In this context it is usually necessary to perform "shift-invert" iterations, which require repeated solution of systems of the form \\ $(A - \sigma M)y = Mx$, (2) \\ for some <em>shift</em> $\sigma$ (which may be near a spectral point) and for various right-hand sides $x$. In large applications systems (2) have to be solved iteratively, requiring <em>"inner iterations"</em>. \\ \\ In this talk we will describe recent progress in the construction, analysis and implementation of fast algorithms for finding such eigenvalues, utilising algebraic domain decomposition techniques for the inner iterations. \\ \\ In the first part we will describe an analysis of inverse iteration techniques for (1) for a model problem in the presence of errors arising from inexact solves of (2). The delicate interplay between the convergence of the (outer) inverse iteration and the choice of tolerance for the inner solves can be used to determine an efficient iterative method provided a good preconditioner for $A$ is available. \\ \\ In the second part we describe an application to the computation of bifurcations in Navier-Stokes problems discretised by mixed finite elements applied to the velocity-pressure formulation. We describe the construction of appropriate preconditioners for the corresponding (3 x 3 block) version of (2). These use additive Schwarz methods and can be applied to any unstructured mesh in 2D or 3D and for any selected elements. An important part of the preconditioner is the adaptive coarsening strategy. At the heart of this are recent extensions of the Bath domain decomposition code DOUG, carried out by Eero Vainikko. \\ \\ An application to the computation of a Hopf bifurcation of planar flow around a cylinder will be given. \\ \\ This is joint work with Jörg Berns-Müller, Andrew Cliffe, Alastair Spence and Eero Vainikko and is supported by EPSRC Grant GR/M59075.
  • Computational Mathematics and Applications Seminar

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