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

As is well known, energy is generally deemed as one of the most important physical invariants in many conservative problems and hence it is of remarkable interest to consider numerical methods which are able to preserve it. In this paper, we are concerned with the energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods. Algebraic conditions in terms of the Butcher coefficients for ensuring the energy preservation, symmetry and quadratic-Casimir preservation respectively are presented. With the presented condition and in use of orthogonal expansion techniques, the construction of energy-preserving integrators is examined. A new class of energy-preserving integrators which is symmetric and of order $2m$ is constructed. Some numerical results are reported to verify our theoretical analysis and show the effectiveness of our new methods.