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Geometric integrators allow preservation of specific geometric properties of the exact flow of differential equation systems, such as energy, phase-space volume, and time-reversal symmetry, and are particularly reliable for long-run integration. In this study, variable step size composition methods and Gauss methods are implemented in Matlab library integrators, and are tested with several representative problems, including the Kepler problem, the outer solar system and a conservative Lotka–Volterra system. Variable step size integrators often perform better than their fixed step size counterparts and the numerical results show excellent long time preservation of the Hamiltonian in these examples.

Sequential Batch Reactors operate in a sequence of steps, where the reactor receives wastewater, treats it, and discharges it. In this paper the operation of the SBR is decomposed into 4 phases, i.e., mixed fill, aerobic reaction, anoxic reaction, and settling. The last phase is described by a detailed partial differential equation model which is solved using the method of lines. The complete model and numerical simulator is then exploited for dynamic optimisation of the SBR operation, and in particular the optimisation of the daily flow subject to constraints on the effluent quality.