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.

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The non-linear equations, when implementing implicit Runge-Kutta methods, may be solved by a modified Newton scheme and by several linear iteration schemes which sacrificed superlinear convergence for reduced linear algebra costs. A linear scheme of this type was proposed, which requires some additional computation in each iteration step. The rate of convergence of this scheme is examined when it is applied to the scalar test problem

Various iterative schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. In one scheme, when applied to an s-stage Runge-Kutta method, each step of the iteration still requires s function evaluations but consists of r(> s) sub-steps. Improved convergence rate was obtained for the case r = s + 1 only. This scheme is investigated here for the case r = ks, k = 2, 3, · · · , and superlinear convergence is obtained in the limit k → ∞ . Some results are obtained for Gauss methods and numerical results are given.