Energetic-Property-Preserving Numerical Schemes for Coupled Natural Systems

Abstract

In this paper, we propose a method for deriving energetic-property-preserving numerical schemes for coupled systems of two given natural systems. We consider the case where the two systems are interconnected by the action−reaction law. Although the derived schemes are based on the discrete gradient method, in the case under consideration, the equation of motion is not of the usual form represented by using the skew-symmetric matrix. Hence, the energetic-property-preserving schemes cannot be obtained by straightforwardly using the discrete gradient method. We show numerical results for two coupled systems as examples; the first system is a combination of the wave equation and the elastic equation, and the second is of the mass−spring system and the elastic equation.

Keywords

• coupled system
• natural system
• energy-preserving numerical scheme
• energy-dissipating numerical scheme
• discrete gradient method
• geometric integration
• port-hamiltonian system