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

Solution of the Optimal Control Problem for the Cahn-Hilliard Equation Using Finite Difference Approximation

This paper is concerned with the designing, analyzing and implementing linear and nonlinear discretization scheme for the distributed optimal control problem (OCP) with the Cahn-Hilliard (CH) equation as constrained. We propose three difference schemes to approximate and investigate the solution behaviour of the OCP for the CH equation. We present the convergence analysis of the proposed discretization. We verify our findings by presenting numerical experiments

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