Structure preserving algorithms with adaptive time step for Birkhoffian systems

Structure preserving algorithms with adaptive time step are systematically developed for Birkhoffian systems. The development mainly consists of construction, implementation, and application of this kind of algorithms. The construction is based on a direct discretization of the Pfaff–Birkhoff principle in which time is treated as a dynamical variable particularly. The resulting discrete Birkhoffian equations then determine a numerical algorithm for iteration which automatically meets the requirement of structure preservation and time step adaptation. Following the construction, an alternative optimization technique of solving discrete Birkhoffian equations and a reasonable method for initialization of the simulation are provided subsequently, for practical implementation of the algorithm. The performance of the developed algorithm is examined finally by preliminary application in typical examples. Numerical results indicate that the time step adaptation leads to a much improved performance on precisely preserving conserved quantities.