On spurious asymptotic numerical solutions of explicit Runge-Kutta methods

Abstract

The bifurcation diagram associated with the logistic equation v" +1 = av"(\-v") is by now well known, as is its equivalence to solving the ordinary differential equation (ODE) u' = crw(l-u) by the explicit Euler difference scheme. It has also been noted by Iserles that other popular difference schemes may not only exhibit period doubling and chaotic phenomena but also possess spurious fixed points. We investigate, both analytically and computationally, Runge-Kutta schemes applied to the equation u'=f\u), for f(u) = <xu{\-u) and f(u) = oat(l-u)(b-u), contrasting their behaviour with the explicit Euler scheme. We determine and provide a local analysis of bifurcations to spurious fixed points and periodic orbits. In particular we show that these may appear below the linearised stability limit of the scheme, and may consequently lead to erroneous computational results.

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.…”

“…It is well known that these discretizing algorithms can cause spurious behavior in the resulting approximate solution map [5], [8], [9], [11], [13], [22], [23], [27]. The study of such spurious behavior and general behavioral differences caused by discretization has come to be known as the dynamics of numerics.…”

“…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].…”

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

“…These bifurcations give us a handle on, respectively, spurious steady solutions, [Burrage and Butcher (1979), Dahlquist (1978), Dekker and Verwer (1984), Lambert (1987) Recent work by Griffiths, Sweby, and Yee (1990) analyzes the existence and stability of spurious steady solutions of explicit (up to 4-stage) Runge-Kutta methods applied to various nonlinear ordinary differential equations (ODEs) with competing equilibria.…”

Nonnormality Effects in a Discretised Nonlinear Reaction-Convection–Diffusion Equation

“…Analysis for autonomous ordinary differential equations can be found in [5,[8][9][10]24]. Other authors [2,4] have with a, b Ͼ 0 constant, f (u) ϭ u(1 Ϫ u), u(x, 0) ϭ (x) added a space dimension and considered the corresponding given, and with boundary conditions specified at x ϭ 0 and reaction-diffusion equations.…”