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Phase space error control for dynamical systems

Higham, D.J. and Humphries, A.R. and Wain, R.J. (2000) Phase space error control for dynamical systems. SIAM Journal on Scientific Computing, 21 (6). pp. 2275-2294. ISSN 1064-8275

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Variable time-stepping algorithms for initial value ordinary differential equations are traditionally designed to solve a problem for a fixed initial condition and over a finite time. It can be shown that these algorithms may perform poorly for long time computations with initial conditions that lie in a small neighborhood of a fixed point. In this regime there are orbits that are bounded in space but unbounded in time, and the classical error-per-step or error-per-unit-step philosophy may be improved upon. A new error criterion is introduced that essentially bounds the truncation error at each step by a fraction of the solution arc length over the corresponding time interval. This new control can be incorporated within a standard algorithm as an additional constraint at negligible additional computational cost. It is shown that this new criterion has a positive effect on the linear stability properties and hence improves behavior in the neighborhood of stable fixed points. Furthermore, spurious fixed points and period two solutions are prevented. The new criterion is shown to be admissible in the sense that it can always be satisfied with nonzero stepsizes. Implementation details and numerical results are given.