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As is well known, energy is generally deemed as one of the most importantphysical invariants in many conservative problems and hence it is of remarkableinterest to consider numerical methods which are able to preserve it. In thispaper, we are concerned with the energy-preserving integration of non-canonicalHamiltonian systems by continuous-stage methods. Algebraic conditions in termsof the Butcher coefficients for ensuring the energy preservation, symmetry andquadratic-Casimir preservation respectively are presented. With the presentedcondition and in use of orthogonal expansion techniques, the construction ofenergy-preserving integrators is examined. A new class of energy-preservingintegrators which is symmetric and of order $2m$ is constructed. Some numericalresults are reported to verify our theoretical analysis and show theeffectiveness of our new methods.

Energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods