The paper is focused on the analysis of parasitism for multivalue numerical methods intended as geometric numerical integrators for Hamiltonian problems. In particular, the main topic is the design of multivalue numerical methods whose parasitic components remain bounded over certain time intervals, opening the path to the development of nearly conservative multivalue methods able to guarantee a control of parasitism in the long time. The analysis of parasitism as well as the development of the corresponding methods is the core of the treatise. The effectiveness of the approach is also confirmed on selected Hamiltonian problems.
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