Towards using feedback mechanisms for numerical integration of Hamiltonian systems: A spring-mass benchmark

Hamiltonian systems are widely used to model dynamics in physics and one of the grand challenges of this type of systems is to obtain numerical simulation results but retaining physical properties in the real physics world, especially for long term numerical time integration. The so-called symplectic algorithms have been proposed to tackle this tough problem but they are quite mathematical and require a deep understanding of symplectic geometry. In this work, we consider an alternative method based on a spring-mass benchmark using feedback mechanisms. By introducing artificial control terms, we are able to cancel the unexpected effects in numerical discretizations which destroy the physical structures of Hamiltonian systems. Numerical simulation shows a great effectiveness of the proposed methodology.