Model Problems in Numerical Stability Theory for Initial Value Problems

Abstract

Abstract. 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 problemsw are studied. Four other categories of equationsmgradient, dissipative, conservative and Hamiltonian systemsmare 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.

Finite element approximation of the Cahn–Hilliard equation on surfaces

“…It is worth noting that Lemma 1 is typical for gradient flows and similar result was proven in [25,40]. The initial condition for u is given by uðx; 0Þ ¼ u 0 ðxÞ; for x 2 S: ð2:8Þ…”

“…The phase separation processes have been successfully investigated with the Cahn-Hilliard equation in a wide variety of non-equilibrium systems. There have been many algorithms and simulations performed using a variety of discretization methods including finite difference, finite volume, finite element and spectral methods, see, e.g., [1,2,8,10,[24][25][26][27]33,35,36,40,[44][45][46] and the references cited therein.…”

Order Stars and Linear Stability Theory

“…However, it is primarily through stability analysis that it is possible to distinguish between different integration techniques; when convergence occurs, it does so for all consistent numerical methods and is distinguished solely by the rate of convergence. A recent review of aspects of stability in the numerical integration over long-time intervals can be found in Stuart and Humphries (1994).…”

Unconditionally stable schemes for higher order inpainting

“…Unconditionally stable and uniquely solvable numerical schemes exist for these equations (cf. [66]). If E(u) is not strictly convex, i.e., λ < 0, multiple minimizers may exist and the gradient flow can possibly expand in u(t).…”