**6. Acknowledgement**

This work is supported by the National Science Foundation (Grant CBET-0709113) and in part by Illinois Clean Coal Institute (Grant 10/ER16).

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**0**

**3** 

**Numerical Integration Techniques Based on a**

**Geometric View and Application to Molecular**

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

In our proposed procedure (Fukuda & Nakamura, 2006), for an ODE defined on an *N*-dimensional phase space Ω we construct an extended phase space Ω of *N* + 1 dimension, by introducing an additional degree of freedom. Then, on Ω we constitute a new ODE, which has an invariant but retains every solution of the original ODE, and we construct efficient integrators for the extended ODE. Advantageous features of our proposal are the simplicity and the applicability to a wide class of ODEs beyond the Hamiltonian equations. In fact, in MD methods, non-Hamiltonian equations are often used (Hoover, 1991); they have been developed (Hoover & Holian, 1996), e.g., to provide more robustness than conventional one or a rapid convergence to a targeted statistical thermodynamic ensemble, or to define a new ensemble itself. Considering such a development, new equations must be designed successively in future studies, and the simplicity and the applicability for the current

Specifically, by the first technique, the invariant can be simply constructed for any (smooth) ODE, including non-Hamiltonian equation. It can thus be easily used to examine the accuracy of numerical integration of the ODE by monitoring the invariant value, as done in

by capturing certain geometric properties of the ODE.

techniques will be useful also in such a circumstance.

**1. Introduction**

**Dynamics Simulations**

*of Physical and Chemical Research)*

*Universités-CNRS*

<sup>1</sup>*Japan* <sup>2</sup>*France*

Ikuo Fukuda1 and Séverine Queyroy2

<sup>1</sup>*Computational Science Research Program, RIKEN (The Institute*

<sup>2</sup>*Laboratoire chimie Provence, chimie théorique, UMR 6264, Aix-Marseille*

