The Essential Stability of Local Error Control for Dynamical Systems

Although most adaptive software for initial value problems is designed with an accuracy requirement—control of the local error—it is frequently observed that stability is imparted by the adaptation. This relationship between local error control and numerical stability is given a firm theoretical underpinning. The dynamics of numerical methods with local error control are studied for three classes of ordinary differential equations: dissipative, contractive, and gradient systems. Dissipative dynamical systems are characterised by having a bounded absorbing set B which all trajectories eventually enter and remain inside. The exponentially contractive problems studied have a unique, globally exponentially attracting equilibrium point and thus they are also dissipative since the absorbing set B may be chosen to be a ball of arbitrarily small radius around the equilibrium point. The gradient systems studied are those for which the set of equilibria comprises isolated points and all trajectories are bounded so that each trajectory converges to an equilibrium point as t → ∞. If the set of equilibria is bounded then the gradient systems are also dissipative. Conditions under which numerical methods with local error control replicate these large-time dynamical features are described. The results are proved without recourse to asymptotic expansions for the truncation error. Standard embedded Runge–Kutta pairs are analysed together with several nonstandard error control strategies. Both error per step and error per unit step strategies are considered. Certain embedded pairs are identified for which the sequence generated can be viewed as coming from a small perturbation of an algebraically stable scheme, with the size of the perturbation proportional to the tolerance τ. Such embedded pairs are defined to be essentially algebraically stable and explicit essentially stable pairs are identified. Conditions on the tolerance τ are identified under which appropriate discrete analogues of the properties of the underlying differential equation may be proved for certain essentially stable embedded pairs. In particular, it is shown that for dissipative problems the discrete dynamical system has an absorbing set B_τ and is hence dissipative. For exponentially contractive problems the radius of B_τ is proved to be proportional to τ. For gradient systems the numerical solution enters and remains in a small ball about one of the equilibria and the radius of the ball is proportional to τ. Thus the local error control mechanisms confer desirable global properties on the numerical solution. It is shown that for error per unit step strategies the conditions on the tolerance τ are independent of initial data while for error per step strategies the conditions are initial-data dependent. Thus error per unit step strategies are considerably more robust

Runge–Kutta Methods for Dissipative and Gradient Dynamical Systems

The numerical approximation of dissipative initial value problems by fixed time-stepping Runge–Kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared. A general class of ordinary differential equations, for which dissipativity is induced through an inner product, is studied throughout. This class arises naturally in many finite dimensional applications (such as the Lorenz equations) and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics. It is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step-size h > 0. The numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor A_h exists; upper-semicontinuity of A_h at h = 0 is established, which shows that, for h small, every point on the numerical attractor is close to a point on the true global attractor A. Under the additional assumption that the problem is globally Lipschitz, it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of A_h at h = 0 is again established. For gradient systems with globally Lipschitz vector fields it is shown that any Runge–Kutta method preserves the gradient structure for h sufficiently small. For general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small. Convergence of the numerical attractor is studied and, for a dissipative gradient system with hyperbolic equilibria, lower semicontinuity at h = 0 is established. Thus, for such a system, A_h converges to A in the Hausdorff metric as h → 0

Model Problems in Numerical Stability Theory for Initial Value Problems

In the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with simple dynamics; this has been motivated by the need to study error propagation mechanisms in stiff problems, a question modeled effectively by contractive linear or nonlinear problems. While this has resulted in a coherent and self-contained body of knowledge, it has never been entirely clear to what extent this theory is relevant for problems exhibiting more complicated dynamics. Recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics. This on-going work is unified and, in particular, striking similarities between this new developing stability theory and the classical linear and nonlinear stability theories are emphasized. The classical theories of A, B and algebraic stability for Runge–Kutta methods are briefly reviewed; the dynamics of solutions within the classes of equations to which these theories apply—linear decay and contractive problems—are studied. Four other categories of equations—gradient, dissipative, conservative and Hamiltonian systems—are considered. Relationships and differences between the possible dynamics in each category, which range from multiple competing equilibria to chaotic solutions, are highlighted. Runge-Kutta schemes that preserve the dynamical structure of the underlying problem are sought, and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given. Algebraic stability, in particular, is seen to play a central role. It should be emphasized that in all cases the class of methods for which a coherent and complete numerical stability theory exists, given a structural assumption on the initial value problem, is often considerably smaller than the class of methods found to be effective in practice. Nonetheless it is arguable that it is valuable to develop such stability theories to provide a firm theoretical framework in which to interpret existing methods and to formulate goals in the construction of new methods. Furthermore, there are indications that the theory of algebraic stability may sometimes be useful in the analysis of error control codes which are not stable in a fixed step implementation; this work is described

Serious Game Evaluation as a Meta-game

Purpose – This paper aims to briefly outline the seamless evaluation approach and its application during an evaluation of ORIENT, a serious game aimed at young adults. Design/methodology/approach – In this paper, the authors detail a unobtrusive, embedded evaluation approach that occurs within the game context, adding value and entertainment to the player experience whilst accumulating useful data for the development team. Findings – The key result from this study was that during the “seamless evaluation” approach, users were unaware that they had been participating in an evaluation, with instruments enhancing rather than detracting from the in-role game experience. Practical implications – This approach, seamless evaluation, was devised in response to player expectations, perspectives and requirements, recognising that in the evaluation of games the whole process of interaction including its evaluation must be enjoyable and fun for the user. Originality/value – Through using seamless evaluation, the authors created an evaluation completely embedded within the “magic circle” of an in-game experience that added value to the user experience whilst also yielding relevant results for the development team

Empirical model for quasi direct current interruption with a convoluted arc

This contribution considers various aspects of a quasi direct current, convoluted arc produced by a magnetic field (B-field) connected in parallel with an RLC circuit that have not been considered in combination. These aspects are the arc current limitation due to the arc convolution, changes in arc resistance due to the B-field and material ablation, and the relative significance of the RLC circuit in producing an artificial current zero. As a result, it has been possible to produce an empirical equation for predicting the current interruption capability in terms of the B-field magnitude and RLC components

Front Solutions for Bistable Differential-Difference Equations with Inhomogeneous Diffusion

This is the published version, also available here: http://dx.doi.org/10.1137/100807156.We consider a bistable differential-difference equation with inhomogeneous diffusion. Employing a piecewise linear nonlinearity, often referred to as McKean's caricature of the cubic, we construct front solutions which correspond, in the case of homogeneous diffusion, to monotone traveling front solutions or, in the case of propagation failure, to stationary front solutions. A general form for these fronts is given for essentially arbitrary inhomogeneous discrete diffusion, and conditions are given for the existence of solutions to the original discrete Nagumo equation. The specific case of one defect is considered in depth, giving a complete understanding of propagation failure and a grasp on changes in wave speed. Insight into the dynamic behavior of these front solutions as a function of the magnitude and relative position of the defects is obtained with the assistance of numerical results

Syntax circuitry: a mobile game for practicing programming language syntax

According to the U.S. Bureau of Labor Statistics [1], computer science professions are among the fastest growing occupations in the U.S., and computer science occupations will add more than half a million new jobs in the next ten years. A similar need for computer professionals is expected in Kazakhstan. Simultaneously, universities in the U.S. and worldwide are seeing poor retention rates in computer science, a major reason being that students often view the early courses in the subject as uninteresting and dull [2]. Game-based learning is one of many techniques proposed to address this issue [2,3,4,5]. We have developed a mobile game that provides an engaging way for students to practice (not learn) the basic syntax of C, C++ and Java (since these languages share the same syntax for basic constructs like declarations, selection and iteration). Learning programming language syntax is a tedious process. Practicing by programming is, of course, ideal, but we believe that a game which is fun to play in their spare time will help students get used to distinguishing correct syntactical constructs quickly