New energy-preserving numerical schemes for Hamiltonian PDEs and formulation of the framework as a discrete mechanics

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In this research, we proposed a new framework for deriving energy-preserving numerical schemes for Hamiltonian partial differential equations. In our framework, energy-preserving schemes are derived by using the symmetry of time translation of the Lagrangian that defines the equation. Since the symmetry used in this framework is not restricted to that of time translation, this method also derives numerical schemes that inherit other conservation laws by using the corresponding symmetries. Extension of this method to systems with holonomic constraints, local discrete conservation laws of the schemes and combination with the finite element exterior calculus were also investigated.