Numerical Integration Techniques Based on a Geometric View and Application to Molecular Dynamics Simulations

In this chapter we address numerical integration techniques of ordinary differential equation (ODE), especially that for molecular dynamics (MD) simulation. Since most of the fundamental equations of motion in MD are represented by nonlinear ODEs with many degrees of freedom, numerical integration becomes essential to solve the equations for analyzing the properties of a target physical system. To enhance the molecular simulation performance, we demonstrate two techniques for numerically integrating the ODE. The first object we present is an invariant function, viz., a conserved quantity along a solution, of a given ODE. The second one is a numerical integrator itself, which numerically solves the ODE by capturing certain geometric properties of the ODE.

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