Runge–Kutta Methods for Dissipative and Gradient Dynamical Systems

Humphries, A. R.; Stuart, Andrew M.

doi:10.1137/0731075

Cited by 95 publications

(63 citation statements)

References 21 publications

(24 reference statements)

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“…For Runge-Kutta methods applied to ordinary differential equations, see [14]. The question of convergence of attractors and of attracting sets for semilinear evolution equations is discussed in, for example, [9], [10], and [16].…”

Section: I(ii) I(s(t)v) T' > 0 and For All V H() I(iii) If I(s(t)v)mentioning

confidence: 99%

The Global Dynamics of Discrete Semilinear Parabolic Equations

Elliott¹,

Stuart²

1993

SIAM J. Numer. Anal.

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Abstract. A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered. The gradient structure of the problem implies that, provided all steady states are isolated, solutions approach a steady state as cx. The dynamical properties of various finite difference and finite element schemes for the equations are analysed. The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods. Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases. However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented. Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep. As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken. To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step. Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep (BDF) methods are studied.

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Section: I(ii) I(s(t)v) T' > 0 and For All V H() I(iii) If I(s(t)v)mentioning

confidence: 99%

The Global Dynamics of Discrete Semilinear Parabolic Equations

Elliott¹,

Stuart²

1993

SIAM J. Numer. Anal.

Self Cite

304

201

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“…As can be seen a complete theory of gradient stability is not yet developed. However, it is woh observing that, if the additional assumption (5.3) (a form of dissipativity) is appended to (4.2) and the equilibria are isolated, then the conclusion of Result 4.4 follows for any algebraically stable RKMsee [36]. 5.…”

Section: !-1mentioning

confidence: 99%

Model Problems in Numerical Stability Theory for Initial Value Problems

Stuart¹,

Humphries²

1994

SIAM Rev.

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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.

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“…Matsuo-Furihata [2] dealt with this problem using dynamical systems theory. Lyapunov-type theorem in discrete dynamical systems (Humphries-Stuart [3]) reveals that if the energy function can serve as a (strong) Lyapunov functional and the level set of the energy function is compact, then the ω-limit set of an arbitrarily chosen initial value is a subset of the fixed points. But their analysis was limited to a specific scalar problem…”

Section: Introductionmentioning

confidence: 99%

An analysis on the asymptotic behavior of multistep linearly implicit schemes for the Duffing equation

2015

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We consider the discrete gradient method for dissipative linear-gradient systems, which strictly replicates the dissipation property, yielding a remarkable stability. However, it also replicates the nonlinearity of an original equation. To overcome this, we can employ multistep linearly implicit schemes as a relaxation; however, it can in turn destroy the originally aimed stability. Matsuo-Furihata (2014) introduced a dynamical systems viewpoint to understand the behavior for a toy scalar problem. In this letter, we show that their method can work also for the two-dimensional Duffing equation. There a new concept of semi-strong Lyapunov functionals is required.

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Runge–Kutta Methods for Dissipative and Gradient Dynamical Systems

Cited by 95 publications

References 21 publications

The Global Dynamics of Discrete Semilinear Parabolic Equations

The Global Dynamics of Discrete Semilinear Parabolic Equations

Model Problems in Numerical Stability Theory for Initial Value Problems

An analysis on the asymptotic behavior of multistep linearly implicit schemes for the Duffing equation

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