Ian Gladwell

Reliable solution of special event location problems for ODEs

Computing the solution of the initial value problem in ordinary differential equations (ODEs) may be only part of a larger task. One such task is finding where an algebraic function of the solution (an event function) has a root (an event occurs). This is a task which is difficult both in theory and in software practice. For certain useful kinds of event functions, it is possible to avoid two fundamental difficulties. It is described how to achieve the reliable solutions of such problems in a way that allows the capability to be grafted onto popular codes for the initial value problem.

Practical aspects of interpolation in Runge-Kutta codes

Runge-Kutta codes solve numerically the initial value problem for a system of ordinary differential equations in a step-by-step manner. The automatically chosen step sizes may vary widely in size depending on the equations, the nature of the error control, and the accuracy desired. Interpolation schemes approximate the solution of the differential equation within the step. Analytical tools are developed and applied to a selection of interpolation schemes with the aim of understanding how the interpolants behave. A novel matter in this context is the preservation of properties inherent in the values and derivatives computed by the Runge-Kutta code. In particular, we show how to preserve local monotonicity so as to achieve a visually pleasing interpolant, and in a way which is especially appropriate to Runge-Kutta codes.

Algorithms for Almost Block Diagonal Linear Systems

Codes for solving systems of ordinary differential equations for use in the method of lines for partial differential equations (PDEs) usually provide only a banded system solver. In this context, a frequently occurring structure is almost block diagonal (ABD). Solving ABD systems by imposing banded structure introduces fill-in and is inefficient. Though robust, efficient ABD software has been developed and used in packages for solving boundary value problems with separated boundary conditions for ordinary differential equations (BVODEs), it has not been generally exploited in PDE software. The situation with bordered almost block diagonal system software for BVODEs with nonseparated boundary conditions is less satisfactory. This survey draws on material from a variety of sources, particularly [P. Amodio et al., Numer. Linear Algebra Appl., {7} (2000), pp. 275--317] and [B. Garrett and I. Gladwell, J. Comput. Methods Sci. Engrg., {1} (2001), pp. 75--98] and the references therein.

Algorithm 670: a Runge-Kutta-Nyström code

A robust, reliable, and efficient code implementing recently developed Runge-Kutta-Nystrom (RKN) formulas is described. Two embedded formula pairs are provided. The lower order pair allows interpolation. The structure of the code is based on a modular approach to the design of software for the numerical solution of ordinary differential equations.

Initial Value Routines in the NAG Library

The hmtorical development of the initial value routines m the NAG library ordinary differential equations chapter ts described. The current contents of the chapter are dmcussed in detail, with emphasis on the overall design of the chapter and on the design of the individual routines in the Runge-Kut ta section. This section and the Adams and backward differentmtion formula sections were all designed to simplify switching between calls to routines m the different sectmns. Within each sectmn there is a main routine with a number of novel interrupt features and also a collection of driver routines which exploit these interrupts. Each of these routines has essentially the same calling sequence as the corresponding routine in the other sections. To achieve this standardization between sections of the chapter, It was necessary to provide an intermediate output facility for the RungeKutta routine. The implementation of this novel facility IS discussed in some detail. Also discussed m a new test used m a Runge-Kutta routine to determine whether a differential system is stiff. The same routine independently calculates global error estimates. Finally, plans for the immediate future development of the chapter are outlined.

Sintering of two particles by surface and grain boundary diffusion – a three-dimensional model and a numerical study

Abstract We outline a model for the sintering of two particles in 3D by surface and grain boundary diffusion, showing the effect of the joining conditions, using a simple formula for grain boundary diffusion. Using centerwise shifts, we implicitly take account of grain boundary migration. We show that whether an evolution will reach an equilibrium state or a quasi-equilibrium state and the kind of equilibrium depends on the joining conditions. The method of lines is used successfully in the computation to exhibit realistic solutions. The model and the computation could be used to simulate the sintering of a cluster of particles by “patching”.

Automatic selection of the initial step size for an ODE solver

Abstract The choice of initial step size is critical for the reliable numerical solution of the initial value problem for a system of ordinary differential equations. Automatic selection of this step size may lead to a more robust and efficient integration than its provision by a user, and is always more convenient. It is especially important for the reliability of an ODE solver used as a module in a larger software package. Previous approaches to making the selection are combined with some new ideas to produce an effective scheme for the automatic choice of the initial step size. Numerical results illustrate the roles played by the individual phases of the algorithm and show that the whole algorithm is both robust and efficient.

Automated selection of mathematical software

Current approaches to recommending mathematical software are qualitative and categorical. These approaches are unsatisfactory when the problem to be solved has features that can “trade-off” in the recommendation process. A quantitative system is proposed that permits tradeoffs and can be built and modified incrementally. This quantitative approach extends other knowledge-engineering techniques in its knowledge representation and aggregation facilities. The system is demonstrated on the domain of ordinary differential equation initial value problems. The results are significantly superior to an existing qualitative (decision tree) system.

Paper: Solving almost block diagonal systems on parallel computers

Abstract Finite difference methods for nonlinear boundary value problems (BVPs) in ordinary differential equations involve solving systems of linear equations at an inner iteration. In the typical case of separated boundary conditions, these systems are almost block diagonal (ABD). A new ‘tearing’ algorithm for the parallel solution of those ABDs is presented here. It is an extension of one proposed by Dongarra and Johnsson [7] for positive definite or strictly diagonally dominant banded systems and it is similar to one proposed by Wright [15] for banded systems. We compare the cost of the new algorithm with costs of other algorithms which might be applied to ABDs. We simulate the use of the algorithm in solving BVPs. The new algorithm is not designed for a specific computer architecture and it is believed that the analysis is sufficiently general to be indicative of performance for most current parallel architectures.

Algorithm 687: a decision tree for the numerical solution of initial value ordinary differential equations

A decision tree to assist in the process of selecting an appropriate algorithm for the numerical solution of initial value ordinary differential equations is described. This initial value tree contains a series of questions that are intended to identify specific features of a user’s problem, relevant to the selection of suitable software. An appropriate basic or general approach for the numerical solution is identified, and then specific sets of codes, incorporating this “generic” approach, are selected. Where possible, the recommended codes are in libraries where the software has been tested extensively and is maintained. The tree can be used with NITPACK (ACM Algorithm 606), a set of software tools that guide a user through a tree in an interactive computer session.