1 July - 31 December 1995

**Organisers**: P Constantin (*Chicago*), JD Gibbon (*Imperial College, London*), JK Hale (*Georgia Tech*), C Sparrow (*Cambridge*). ASI Organiser: P Glendinning (*Cambridge*)

**Background**

This small workshop will take place as part of the Newton Institute programme, `From Finite to Infinite Dimensional Dynamical Systems' The general aim of this programme is to bring together researchers with an analytic training (in partial differential equations, for example) and those with a more topological approach in an attempt to understand `large' systems.

Numerical studies of dynamical systems require high quality results to secure insight into the underlying processes and structure. Very accurate numerical methods are required to achieve precise results and sophisticated error control methods are needed to establish reliability of answers. Numerical analysts will report on their research into improving accuracy and error control for systems of ODEs or PDEs.

**10:00**C M Elliott, Sussex, UK*Numerical Solution of a mean field model in super-conductivity coupling a hyperbolic equation to an elliptic equation***11:00***COFFEE***11:30**P K Jimack, Leeds, UK*Stability of the Moving Finite Element Method for a Class of Parabolic Partial Differential Equations***12:30***LUNCH***2:00**M Berzins, Leeds, UK*Spatio-Temporal Error Control for time-dependent PDEs***3:00**J Norbury, Oxford, UK*Some Math Problems in Weather Prediction***4:00***TEA***4:30**A T Hill, Bath, UK*The inheritance of dissipativity by linear multistep and Runge-Kutta methods*

**10:00**M Baines, Reading, UK*Multidimensional Upwinding and Grid Adaption***11:00***COFFEE***11:30**E Suli, Oxford, UK*Posteriori Error Analysis and adaptivity for time-dependent problems***12:30***LUNCH***2:00**J Butcher Auckland,NZ & IRISA, FR*Runge-Kutta methods: a century of accurate computations***3:00**R W Wright, Imperial College, London, UK*Continuation for Singularly Perturbed Two-Point Boundary Value Problems***4:00***TEA***4:30**D R Moore, Imperial College, London, UK*Accurate Interpolants for the solutions of ODE systems*

The workshop will take place in the Newton Institute's purpose-designed building, in a pleasant area in the west of Cambridge, about one mile from the centre of the City. The Newton Institute can provide assistance with finding local accommodation - the cost of which is likely to be 25 - 40 pounds per day including breakfast. There will be a small registration fee to cover the cost of lunch. There may be some financial help available to support UK participants.

To register, please obtain and return a **Registration Form** from/to Mike Sekulla at

Isaac Newton Institute,

20 Clarkson Road,

Cambridge CB3 0EH.

**Tel** (44) 1223-335984

**Fax**(44) 1223-330508

Further information about the workshop programme may be obtained from Dan Moore and Colin Sparrow

Department of Mathematics, University of Reading, Reading RG6 2AX, UK.

*Multidimensional Upwinding and Grid Adaption*

Upwind methods have long been popular in the modelling of 1-D compressible flows with discontinuities governed by the Euler equations. They have also been frequently used to solve these equations in higher dimensions via operator splitting. However these techniques remain essentially one-dimensional and can be inadequate for capturing flow features such as shocks or shears not aligned with the grid.

The last ten years has seen a number of attempts to rectify this state of affairs by introducing multidimensional physics into upwind methods. Three building blocks are required for these methods: a wave decomposition model, a way to achieve conservation, and a compact advection scheme. We review recent advances in each of these areas and present results which show that these methods are now achieving their goal.

The upwind advection schemes can easily be combined with grid movement techniques. A simple approach is described which gives excellent resolution of discontinuities in steady advection problems as well as in the simulation of complicated compressible flows.

School of Computer Studies, University of Leeds, Leeds LS2 9JT, UK.

*Spatio-Temporal Error Control for time-dependent PDEs.*

This talk is concerned with the issue of spatial and temporal errors in the numerical solution of time dependent p.d.e.s . A general error equation will be used to describe a number of approaches adopted in this area. The issues of how to estimate the spatial error will be considered and a short study of the introduction of the error generated by discrete remeshing presented. Examples from adaptive codes will be used to illustrate the main points of the talk.

Department of Mathematics, The University of Aukland, Provate Bag 92019, Aukland, NZ.

*Runge-Kutta methods: a century of accurate computations*

The idea in Runge's famous 1895 paper was to make a modest improvement over the low accuracy of the traditional Euler method. This was taken much further in Kutta's paper and in more recent work. Today methods with orders 8 and higher are available. The quest for increasingly high orders, and therefore greater efficiency in the performance of accurate computations, will be surveyed. The order concept itself will be discussed and the conditions for order expressed in a simple form. This enables an analysis to be performed of what orders are actually possible, and at the same time gives insight into how accurate methods can be constructed. In the early days of the Runge-Kutta method, hand computations were performed and monitoring of the behaviour of the solution was a matter of observation and judgement. Today everything has to be done automatically and error estimation and stepsize adjustment are built into the algorithm. The requirements for the enhancements of the basic numerical methods to make these automatic features possible will be discussed as will the provision of so-called dense output. The introduction of implicit Runge-Kutta methods presented new opportunities and challenges. It was soon recognised that these new methods have their natural role in the solution of so-called stiff problems, where strong stability is turned into a severe handicap for traditional explicit methods. Some of the properties and some of the apparent limitations of implicit Runge-Kutta methods will be discussed and their role in accurate computations will be assessed.

Centre for Mathematical Analysis and its Applications, University of Sussex, UK.

*Numerical Solution of a mean field model in super-conductivity coupling a hyperbolic equation to an elliptic equation.*

School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.

*The inheritance of disapativity by linear multistep and Runge-Kutta methods*

Suppose that you want a picture of the asymptotic dynamics of a dissipative ODE or PDE, but don't have a priori information of the location of structures within the attractor. In this case it is necessary to compute a number of trajectories over long time intervals. With the exception of a few special cases, error control over an unbounded time interval is impossible. So, the most that can be hoped for is that the computed orbits will, in some sense, be representative of the long-time behaviour of the underlying system.

If your numerical method is dissipative, its trajectories are guaranteed to eventually lie in a bounded neighbourhood of the true attractor. In this case, one can discuss the attractor of the numerical method and its convergence --- i.e. whether the asymptotic trajectories of the numerical method will be close to those of the underlying solution, for sufficiently small values of the discretisation paramter.

In this talk, I report work on classifying which common methods are dissipative.

School of Computer Studies, University of Leeds, Leeds LS2 9JT, UK.

*Stability of the Moving Finite Element Method for a Class of Parabolic Partial Differential Equations.*

Department of Mathematics, Imperial College, London SW7 2BZ, UK.

*Accurate Interpolants for the solution of ODE systems*

Modern accurate methods for Ordinary Differential Equations can generate very accurate answers on quite coarse grids. Subsequent analysis of these results may require an accurate interpolation of the solution between these widely spaced points. Although high order polynomial interpolants can be constructed to fit value and derivative data over several mesh points, non-local schemes can become inaccurate near boundaries.

Following a suggestion of John Butcher (private communication, 1995), compact schemes have been devised to construct 6th and 8th order accurate interpolants over a single mesh interval. These schemes fit an interpolant to symmetric internal points and express the relationship between the function value and its derivatives there. These relationships may be solved iteratively or as a limited set of non-linear equations to find internal function values. These values may be used with the interval end data to construct the accuate polynomial interpolant.

With two interior fitting points, a 6th order accurate family of interpolants can be found. With four interior points an 8th order accurate family can be constructed. The interior points can be chosen for convenience, to minimize the error function or to create an interpolant with optimal integral properties.

For systems of second order differential equations, an accurate interpolant can be constructed by these methods can be found using just the midpoint of the mesh interval.

Mathematics Institute, University of Oxford, Oxford ,UK.

*Some Math Problems in Weather Prediction*

Computing Laboratory, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK.

*A posteriori error analysis and adaptivity for time-dependent problems*

The aim of this talk is to present new a posteriori error bounds for finite element approximations of hyperbolic partial differential equations and parabolic equations with dominant hyperbolic behaviour. The bounds exploit the computable finite element residual. We shall demonstrate, through numerical experiments, the potentials of the approach as well as the implementation of the theoretical results into space/time adaptive finite element algorithms.

Department of Mathematics, Imperial College, London SW7 2BZ, UK.

*Continuation for Singularly Perturbed Two-Point Boundary Value Problems*

The efficient numerical solution of stiff two-point boundary value problems has proved to be a difficult and elusive task. The central difficulties with such problems arise due to the fact that it is often very hard to determine both a mesh that will yield a sufficiently accurate numerical solution, and an initial guess to the solution which will lead to the convergence of Newton's method. One way of overcoming these problems is to use continuation. In this approach a sequence of progressively more difficult problems is solved by using information from one problem in the sequence to solve for the next. In this talk a particular continuation algorithm for stiff two-point boundary value problems is described and it is shown how this can lead to enormous gains in efficiency for extremely difficult problems.