Compensating for order variation in mesh refinement for direct transcription methods. II: Computational experience
A typical problem in the integration of differential algebraic equations is that numerical methods do not show the expected order but a lower one. In Part I [ibid. 125, No.~1-2, 147-158 (2000; Zbl 0971.65075)] the authors described a new approach for dealing with this effect in the context of the direct transcription approach to solving optimal control problems. It has now been implemented in the industrial optimisation code SCOS and the present article discusses the performance of the new approach compared with an older version of SCOS. The authors discuss six concrete test problems. The ``alp rider'' is a typical example of a system trying to follow a surface which may change abruptly. The second problem concerns a close flyby of Venus and contains rapid transients. The third benchmark comes from chemical engineering and contains large constants so that it is difficult to obtain highly accurate solutions. The fourth problem contains a constraint that is active only half of the time so that significant variations in the order appear. Furthermore, there are corners where smoothness is lost and again rapid transients. In the fifth problem an index three constraint becomes active and the control is not always uniquely determined by the constraint. Finally, a classical benchmark from multibody dynamics is studied: ``Andrew's squeezer''. The authors report the following observations. On simple problems the old and the new approach perform comparably. On problems with rapid transients or with state constraints, the new approach usually outperforms the old one, sometimes even drastically. The produced meshes are smaller, CPU times lower and intermediate iterates often provide better approximations. The main reason lies in the fact that the new approach better locates critical areas and refines the mesh accordingly.
scientific article; zbMATH DE number 1787639