Energetic-Property-Preserving Numerical Schemes for Coupled Natural Systems

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.

The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations

In this paper we focus on the analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations. A novel energy-preserving scheme is developed based on the discrete gradient method and the Duhamel principle. The local error, global convergence and nonlinear stability of the new scheme are analysed in detail. Numerical experiments are implemented to compare with existing numerical methods in the literature, and the numerical results show the remarkable efficiency of the new energy-preserving scheme presented in this paper.

An Effective Genetic-Based Mathematical Computation Algorithm for Minimizing the Lennard-Jones Potential

In molecular dynamics, molecular mechanics and quantum mechanics for a complicated system, most of the numerical calculations are at the Lennard-Jones potential. So, effective treatment of Lennard-Jones potential shows an important role. This paper presents an effective algorithm for the optimization of Lennard-Jones potential. The algorithm is a genetic-based algorithm associated with the owerful discrete gradient method and the correct choice of initial coordinates. Numerical results show the effectiveness of the present algorithm and new bestresults for 39,40,42,48,55,75,76,97 atoms were found. These results might be used to study the structure of spherical viruses and molecules.