Comlab

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.

2.00 pm Professor Iain Duff (RAL) Opening remarks

2.15 pm Professor M J D Powell (University of Cambridge)

Some developments of work with Alan on cubic splines

3.00 pm Professor Kevin Burrage (University of Queensland)

Stochastic models and simulations for chemically reacting systems

3.30 pm Tea/Coffee

4.00 pm Professor John Reid (RAL)

Sparse matrix research at Harwell and the Rutherford Appleton Laboratory

4.30 pm Dr Ian Jones (AEA PLC)

Computational fluid dynamics and the role of stiff solvers

5.00 pm Dr Lawrence Daniels (Hyprotech UK Ltd)

Current work with Alan on ODE solvers for HSL

Long-step primal-dual path-following algorithms constitute the

framework of practical interior point methods for

solving linear programming problems. We consider

such an algorithm and a second order variant of it.

We address the problem of the convergence of

the sequences of iterates generated by the two algorithms

to the analytic centre of the optimal primal-dual set.

Several real Lie and Jordan algebras, along with their associated

automorphism groups, can be elegantly expressed in the quaternion tensor

algebra. The resulting insight into structured matrices leads to a class

of simple Jacobi algorithms for the corresponding $n \times n$ structured

eigenproblems. These algorithms have many desirable properties, including

parallelizability, ease of implementation, and strong stability.

The convergence of Krylov subspace methods like conjugate gradients

depends on the eigenvalues of the underlying matrix. In many cases

the exact location of the eigenvalues is unknown, but one has some

information about the distribution of eigenvalues in an asymptotic

sense. This could be the case for linear systems arising from a

discretization of a PDE. The asymptotic behavior then takes place

when the meshsize tends to zero.

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We discuss two possible approaches to study the convergence of

conjugate gradients based on such information.

The first approach is based on a straightforward idea to estimate

the condition number. This method is illustrated by means of a

comparison of preconditioning techniques.

The second approach takes into account the full asymptotic

spectrum. It gives a bound on the asymptotic convergence factor

which explains the superlinear convergence observed in many situations.

This method is mathematically more involved since it deals with

potential theory. I will explain the basic ideas.

We develop an algorithm for estimating the local Sobolev regularity index

of a given function by monitoring the decay rate of its Legendre expansion

coefficients. On the basis of these local regularities, we design and

implement an hp--adaptive finite element method based on employing

discontinuous piecewise polynomials, for the approximation of nonlinear

systems of hyperbolic conservation laws. The performance of the proposed

adaptive strategy is demonstrated numerically.

The study of the finite precision behaviour of numerical algorithms dates back at least as far as Turing and Wilkinson in the 1940s. At the start of the 21st century, this area of research is still very active.

We focus on some topics of current interest, describing recent developments and trends and pointing out future research directions. The talk will be accessible to those who are not specialists in numerical analysis.

Specific topics intended to be addressed include

• Floating point arithmetic: correctly rounded elementary functions, and the fused multiply-add operation.
• The use of extra precision for key parts of a computation: iterative refinement in fixed and mixed precision.
• Gaussian elimination with rook pivoting and new error bounds for Gaussian elimination.
• Automatic error analysis.
• Application and analysis of hyperbolic transformations.

We describe "avoidance of crossing" and its explanation by von

Neumann and Wigner. We show Lax's criterion for degeneracy and then

discover matrices whose determinants give the discriminant of the

given matrix. This yields a simple proof of the bound given by

Ilyushechkin on the number of terms in the expansion of the discriminant

as a sum of squares. We discuss the 3 x 3 case in detail.