In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid {t n } N n=0 can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., t n = Φ(τ n ), where τ n = n/N and the map Φ is monotonically increasing with Φ(0) = 0 and Φ(1) = 1. The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines Φ, and a tolerance requirement which determines N . Given any strongly stable multistep method, there is an N * such that the method is zero stable for N > N * , provided that Φ ∈ C 2 [0, 1]. Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy h n /h n−1 = 1 + O(N −1 ) as N → ∞. The results are exemplified for BDF-type methods.

In this paper we investigate the numerical solution of non-linear equations in an abstract (Banach space) setting. The main result is that the convergence can be guaranteed by two, directly checkable conditions (namely, by the consistency and the stability). We show that these conditions together are a sufficient, but not necessary condition for the convergence. Our theoretical results are demonstrated on the numerical solution of a Cauchy problem for ordinary differential equations by means of the explicit Euler method.

In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, chp+ 1y(p+ 1)(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.

In this paper we investigate the N-stability notion in an abstract Banach space setting. The main result is that thanks to this notion we have an alternative opportunity for verifying the stability of the numerical solution for periodic initial-value reaction-diffusion problems.

Abstract:In this paper we investigate the T-stability of one-step methods for initial-value problems. The main result is that we extend the classical result (the well-known Euler method) for variable step size explicit and implicit one-step methods. In addition, we give further properties for the theory of T-stability of nonlinear equations in an abstract (Banach space) setting.