Conference PaperPDF Available

The dynamics of some linear multistep methods with step-size control

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... A transient phase as the solution relaxes to its rest state. Insight into the dynamical behavior in this phase can be obtained through a modified equation approach (see [29], [10], [9]). We reparameterize time by a pseudo-time variable s discretized with constant step-size ∆s = (12ε) 1/3 . ...
... For a 256-element uniform mesh, the above remarks are validated by the computational results shown in Figure 3.2 where we show both the analytical solution 9 Copyright (C) Maplesoft, a division of Waterloo Maple Inc. 10 For 0 < t < 10 −2 , u − u erf ∞ < 10 −12 , and we can ensure comparable accuracy at later times u − un ∞ < 10 −12 by taking n = 5/π √ νt terms of the FS. (solid line) and the FE solution (broken lines) at four values of t. ...
Article
Even the simplest advection-diffusion problems can exhibit multiple time scales.This means that robust variable step time integrators are a prerequisite if such problems are tobe efficiently solved computationally. The performance of the second order trapezoid rule usingan explicit Adams–Bashforth method for error control is assessed in this work. This combination isparticularly well suited to long time integration of advection-dominated problems. Herein it is shownthat a stabilized implementation of the trapezoid rule leads to a very effective integrator in othersituations: specifically diffusion problems with rough initial data; and general advection-diffusionproblems with different physical time scales governing the system evolutio (2) Adaptive time-stepping for incompressible flow. Part I: Scalar advection-diffusion | Request PDF. Available from: https://www.researchgate.net/publication/313665602_Adaptive_time-stepping_for_incompressible_flow_Part_I_Scalar_advection-diffusion [accessed Oct 07 2018].
... Adaptivity is widely used in the solution of ordinary differential equations (ODEs) in an attempt to optimize effort expended per unit of accuracy. The adaptation strategy can be viewed heuristically as a fixed time-step algorithm applied to a time re-scaled differential equation [5] and it is of interest to study convergence of the algorithms as the tolerance employed to control adaptation is reduced to zero [14]. However adaptation also confers stability on algorithms constructed from explicit time-integrators, resulting in better qualitative behavior than for fixed time-step counter-parts. ...
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The ergodicity of adaptive time-stepping algorithms for stochastic differential equations is studied. It is assumed that the noise is non-degenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition; the equation is then geometrically ergodic (converges to statistical equilibrium exponentially quickly). However, if the drift is not linearly bounded then explicit fixed time-step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work it is shown that simple adaptive time-stepping strategies can cure this problem. A particular scheme is analyzed in some detail, and proved ergodic. Several others are studied numerically and the conclusions of the analysis thereby extended. In addition to proving ergodicity, we also prove a law of the iterated logarithm and exponential moment bounds, both generalizing results known to hold for the SDE itself. Collectively the results in this paper form stability results for the adaptive SDE methods studied.
... In a notable paper, Hall [30] established a remarkable connection between accuracy and stability for error control schemes. We illustrate this with a simple example modified from [30] and [26]: consider (1.1) with p = 1 and f(u) = -u. ...
Article
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.
... This approach can be generalised to cope with period 2 solutions; see [14]. The dynamics of variable timestepping algorithms are analysed in [5] 2. Motivation for the methods and results. In this section we present two examples which illustrate the effect of spurious period 2 solutions on the dynamics of discretisations. ...
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The dynamics of the theta method for arbitrary systems of nonlinear ordinary differential equations are analysed. Two scalar examples are presented to demonstrate the importance of spurious solutions in determining the dynamics of discretisations. A general system of differential does not generate spurious solutions equations is then considered. It is shown that the choice 0 of period 2 in the timestep n. Using bifurcation theory, it is shown that for 0 1 / 2 the theta method does generate spurious solutions of period 2. The existence and form of spurious solutions are examined in the limit At 0. The existence of spurious steady solutions in a predictor-corrector method is proved to be equivalent to the existence of spurious period 2 solutions in the Euler method. The theory is applied to several examples from nonlinear parabolic equations. Numerical continuation is used to trace out the spurious solutions as At is varied. Timestepping experiments are presented to demonstrate the effect of the spurious solutions on the dynamics and some complementary theoretical results are proved. In particular, the linear stability restriction At/Ax 2 <_ 1 / 2 for the Euler method applied to the heat equation is generalised to cope with a nonlinear problem. This naturally introduces a restriction on At in terms of the initial data; this restriction is necessary to avoid the effect of spurious periodic solutions.
... Other investigations of mean-square stability with fixed time-steps include [2, 3, 22, 28, 1, 24, 29, 8, 30]. For adaptive time-stepping ODE solvers, it has long been known that adaptivity based upon local error control(s) can impart desirable stability properties for linear and nonlinear equations [12, 11], especially for 'stiff' problems, even when the underlying method is explicit and has a small stability region. This occurs because a consequence of the error control is to force the time-steps close to, but below, the linear stability limit. ...
Article
We consider stability properties of a class of adaptive time-stepping schemes based upon the Milstein method for stochastic differential equations with a single scalar forcing. In particular, we focus upon mean-square stability for a class of linear test problems with multiplicative noise. We demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls, one for the drift term and one for the diffusion. (c) 2005 Elsevier B.V. All rights reserved.
... A transient phase as the solution relaxes to its rest state. Insight into the dynamical behavior in this phase can be obtained through a modified equation approach (see [29], [10], [9]). We reparameterize time by a pseudo-time variable s discretized with constant step-size ∆s = (12ε) 1/3 . ...
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Even the simplest advection-diffusion problems can exhibit multiple time scales. This means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally. The performance of the second order Trapezoid Rule using an explicit Adams-Bashforth method for error control is assessed in this work. This combination is particularly well suited to long time integration of advection-dominated problems. Herein it is shown that a stabilized implementation of the Trapezoid Rule leads to a very effective integrator in other situations: specifically diffusion problems with rough initial data; and general advection-diffusion problems with different physical time scales governing the system evolution.
Article
We consider a phase space stability error control for numerical simulation of dynamical systems. We illustrate how variable time-stepping algorithms perform poorly for long time computations which pass close to a fixed point. A new error control was introduced in [9], which is a generalization of the error control first proposed in [8]. In this error control, the local truncation error at each step is bounded by a fraction of the solution arc length over the corresponding time interval. We show how this error control can be thought of either a phase space or a stability error control. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In particular, we analyze the forward Euler method applied to the linear system whose coefficient matrix has real and negative eigenvalues. We also consider the dynamics in the neighborhood of saddle points. We introduce a step-size selection scheme which allows this error control to be incorporated within the standard adaptive algorithm as an extra constraint at negligible extra computational cost. Theoretical and numerical results are presented to illustrate the behavior of this error control. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Article
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 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 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 Bτ is hence dissipative. For exponentially contractive problems the radius of Bτ 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.
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It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.
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