One-leg Methods Research Articles

Stability of Two-Step Methods for Variable Integration Steps

Two of the most commonly used methods, the trapezoidal rule and the two-step backward differentiation method, both have drawbacks when applied to difficult stiff problems. The trapezoidal rule does not sufficiently damp the stiff components and the backward differentiation method is unstable for certain stable variable-coefficient problems with variable-steps. In this paper we show that there exists a one-parameter family of two-step, second-order one-leg methods which are stable for any dissipative nonlinear system and for any test problem of the form $\dot x = \lambda (t)x$, $\operatorname{Re} \lambda (t) \leq 0$, using arbitrary step sequences.

On Contractive Linear Multistep and One-Leg Methods

Contractive variable-formula methods for the integration of y'=lambda(t)y (lambda(t)) restricted to some region of interest) are formed by using polyhedral norms for showing the contractivity of several one-leg or linear multistep formulas. When possible the results are given for variable step-size.

Non-linear stability of a general class of differential equation methods

For a class of methods sufficiently general as to include linear multistep and Runge-Kutta methods as special cases, a concept known as algebraic stability is defined. This property is based on a non-linear test problem and extends existing results on Runge-Kutta methods and on linear multistep and one-leg methods. The algebraic stability properties of a number of particular methods in these families are studied and a generalization is made which enables estimates of error growth to be provided for certain classes of methods.

Multistep and one-leg methods for implicit mixed differential algebraic systems

General one-leg and multistep methods for mixed systems of implicit differential equations and algebraic constraints are defined. Such systems are encountered frequently in circuit analysis. Three different implementations of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A (\alpha)</tex> -contractive two-step Adams method are shown to be second-order accurate by an analysis of the local truncation errors. These methods are tested numerically against the familiar two-step second-order backward differentiation method.