A Unified Approach to Spurious Solutions Introduced by Time Discretisation. Part I: Basic Theory

Abstract

Abstract. The asymptotic states of numerical methods for initial value problems are examined.In particular, spurious steady solutions, solutions with period 2 in the timestep, and spurious invariant curves are studied. A numerical method is considered as a dynamical system parameterised by the timestep h. It is shown that the three kinds of spurious solutions can bifurcate from genuine steady solutions of the numerical method (which are inherited from the differential equation) as h is varied. Conditions under which these bifurcations occur are derived for Runge-Kutta schemes, linear multistep methods, and a class of predictor-corrector methods in a PE(CE) M implementation. The results are used to provide a unifying framework to various scattered results on spurious solutions which already exist in the literature. Furthermore, the implications for choice of numerical scheme are studied. In numerical simulation it is desirable to minimise the effect of spurious solutions.Classes of methods with desirable dynamical properties are described and evaluated.

Transformation of Local Bifurcations Under Collocation Methods

“…We also wish to discover the defining conditions for any spurious bifurcation behavior. This work differs somewhat from that reported in Iserles et al [14] and Griffiths et al [5], in which the ODEs under discussion were not parameter-dependent and all spurious bifurcations resulted from variation of h.…”

“…Runge-Kutta and predictor-corrector schemes generally possess spurious fixed points for nonlinear problems, but this is not the case with linear multistep methods [14]. A number of authors have studied the properties of spurious fixed points.…”

“…Such behavioral differences typically take the form of spurious fixed points and bifurcations to spurious asymptotes [5], [11], [14]. Runge-Kutta and predictor-corrector schemes generally possess spurious fixed points for nonlinear problems, but this is not the case with linear multistep methods [14].…”

Numerical Implementation of Gradient Algorithms

“…Arguably the fact that the ODE is explicitly written in the linear-gradient form could have been taking as the starting assumption for the studied systems, however it is worth emphasizing that the linear-gradient form given by Equation (1) is not unique [10]. An ODE that has been rewritten in the form of linear gradient, as in Equation (1), can be discretized by considering the following integration method:…”

Upper Semicontinuity of Attractors for Linear Multistep Methods Approximating Sectorial Evolution Equations

“…Examples of this phenomenon are considered, for instance, in the papers of Griffiths and An appreciation of the difficulties identified in the last paragraph has led to efforts, principally in the context of ODEs, to identify particular structures in the attractor, such as fixed points and periodic orbits, and to test whether such structures are preserved by common time discretizations, such as Runge-Kutta and linear multistep methods. This work has been developed by Iserles et al in the articles [10], [20][21][22]. The inheritance of stability properties for periodic orbits has also been studied by investigating the circumstances under which trajectories are approximated in a C1 sense.…”