Analysis of the dynamics of local error control via a piecewise continuous residual |
| |
Authors: | D J Higham A M Stuart |
| |
Institution: | (1) Department of Mathematics, University of Strathclyde, G1 1XH Glasgow, Scotland;(2) Scientific Computing and Computational Mathematics Program Division of Mechanics and Computation, Stanford University, 94305-4040 Stanford, CA, USA |
| |
Abstract: | Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations.
The results are cast in terms of the local error tolerance. Under theassumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution
exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known
to hold for thematlab ode23 algorithm 10] when applied to a variety of problems.
Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution
as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete
analogs of the long-time dynamical properties of the equation—dissipative, contractive and gradient systems are analysed in
this way.
Supported by the Engineering and Physical Sciences Research Council under grants GR/H94634 and GR/K80228.
Supported by the Office of Naval Research under grant N00014-92-J-1876 and by the National Science Foundation under grant
DMS-9201727. |
| |
Keywords: | 34C35 34D05 65L07 65L20 65L50 |
本文献已被 SpringerLink 等数据库收录! |
|