Stable Multistep Methods Research Articles

On the zero-stability of multistep methods on smooth nonuniform grids

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=0}^N can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., t_n = {varPhi }(tau _n), where tau _n = n/N and the map {varPhi } is monotonically increasing with {varPhi }(0)=0 and {varPhi }(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 {varPhi }, 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 {varPhi }in C^2[0,1]. Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy h_n/h_{n-1} = 1 + {mathrm {O}}(N^{-1}) as Nrightarrow infty . The results are exemplified for BDF-type methods.

On a multistep method approximating a linear sectorial evolution equation

This paper concerns the linear multistep approximation of an abstract dissipative linear sectorial evolution equation on a Banach space X . We study how well the semigroup generated by a sectorial operator A is approximated by the numerical semigroup generated by a q-step, strictly A(α)-stable multistep method. An optimal order error bound is obtained.

Stable multistep methods of numerical integration of rigid systems of differential equations

We consider the basic propositions of the theory of multistep fractional-rational numerical methods with a variable step of integration. We establish general regularities in determining the coefficients of the methods. We prove the A-stability of these methods of arbitrary order, and also the absence of extraneous roots of the characteristic equations of the methods.

High order adaptive methods of Nyström-Cowell type

A class of unconditionally stable multistep methods is discussed for solving initial-value problems of second-order differential equations which have periodic or quasiperiodic solutions. This situation frequently occurs in celestial mechanics, in nonlinear oscillations and various other situations. The methods depend upon a parameter ω > 0, and integrate exactly trigonometric functions along with algebraic polynomials. In this paper we show a procedure for the construction of adaptive Nyström-Cowell formulas of arbitrarily high order of accuracy, and reduce to the classical Nyström-Cowell methods for ω = 0. Our methods compare advantageously with other methods.

On the convergence of multistep methods for nonlinear stiff differential equations

Convergence estimates are given forA(?)-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturbation arguments with frequency domain techniques.

On the convergence of multistep methods for nonlinear stiff differential equations

Convergence estimates are given for A ( )-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturba...

A relationship between order and degree for a stable formula with advanced nodes

A major problem in the implementation of k-step methods is to determine their accuracy. This problem is solved using relationships between order and degree for a k-step method. A relationship between order and degree is found for stable multistep methods with a second derivative.

A note on multistep methods and attracting sets of dynamical systems

We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.

Converging multistep methods for initial value problems involving multivalued maps

Every consistent and strongly stable multistep method of stepnumberk yields a solution, of the setvalued initial value problem\(\dot y \in F(t,y),y(t_0 ) = y_0 \). The setF(t, z) is assumed to be nonvoid, convex and closed. Upper semicontinuity of F with respect to both variables is not required everywhere. If the initial value problem is uniquely solvable, the solutions of the multistep method will converge to the solution of the continuous problem. These results carry over to functional differential equations\(\dot y \in F(t,M_t y)\) of Volterra type and to discontinuous problems\(\dot y(t) = f(t,M_t y)\) in the sense of A.F. Filippov. A difference method is applied to the discontinuous delay equation\(\ddot x(t) + 2D\dot x(t) + \omega ^2 x(t) = = - \operatorname{sgn} (x(t - \tau ) + \dot x(t - \tau ))\). In the limit τ→0 we obtain results for the problem\(\ddot x + 2D\dot x + \omega ^2 x = = - \operatorname{sgn} (x + \dot x)\) which cannot be solved classically everywhere.

Orbitally stable multistep methods

Lambert and Watson (1976) examine the family of symmetric linear multistep methods for the special second-order initial value problem, and connect the property of symmetry with a property of periodicity. The problems of celestial mechanics may be formulated as second-order initial value problems, but these frequently incorporate the first derivative explicity. It is common for such equations to be reduced to a system of first-order equations. Thus motivated, we utilize ideas from the aforementioned paper to determine the family of linear multistep methods for first-order initial value problems that possess an analogous property of periodicity. This family of orbitally stable methods is illustrated by examining the regularized equations of motion of an artificial earth satellite in an oblate atmosphere.

Stiff Stability and Its Relation to $A_0 $- and $A(0)$-Stability

Necessary and sufficient conditions for an $A_0 $-stable multistep method to be stiffly stable or $A(0)$-stable respectively, are established. As an application of these results one finds that Cryer’s k-step method [1] is $A(0)$- and stiffly stable. Hence there exist $A(0)$- and stiffly stable k-step methods of order k. Moreover it is shown that to any stiff stability parameter $D > 0$ there exists a stiffly stable k-step method of order k. Plots of the stability region of Cryer’s method for $k = 1,2, \cdots ,7$ are given. The angle $\alpha $ of $A(\alpha )$-stability and D of stiff stability of Cryer’s method are listed for $k = 1,2, \cdots ,16$.

Highly Stable Multistep Methods for Retarded Differential Equations

The simplest initial value problem for delay differential equations takes the form \[ {\text{(1)}}\qquad \dot y(t) = - \lambda y(t - \tau ),\quad t > 0;\qquad y(t) = \psi (t),\quad - \tau \leqq t \leqq 0.\] If the step size h is chosen so that $\tau = mh$ where m is an integer, then the linear multistep method $\{ \rho ,\sigma \} $ may be applied to (1) to yield an approximate solution $y^h $ defined at the gridpoints $t_j = jh$.It is known that $y(t) \to 0$ as $t \to \infty $ for all initial data $\psi $ iff $\lambda \in (0,{\pi / 2})$. In analogy with the concept of A-stability, the method $\{ \rho ,\sigma \} $ is called $DA_0 $-stable if $y^h (t_j ) \to 0$ as $j \to \infty $ for all initial data $\psi $ and all positive integers m. The properties of $DA_0 $-stable methods are studied.