Runge_Kutta repository

Purpose

This project hosts programs demonstrating two integration algorithms used in molecular dynamics : velocity-Verlet and Runge-Kutta to forth-order (RK4). I developed those codes during my PhD thesis using the code developed by Jofre. The context is that of molecular dynamics simulation of trapped of ions in electric fields.

Codes

Several codes are presented in this repository. Two codes are intended to demonstrate integration algorithms in the case of a single particle in 1D. A more advanced program is dedicated to the molecular dynamics of $N>1$ particles in a mix of radio-frequency and static fields, with Coulomb interaction and friction, using RK4.

1D codes

Verlet_Pendulum.ipynb is a short Python code demonstrating the velocity-Verlet algorithm for a simple pendulum. Created with jupyter notebook and Python 3.7.

Runge_Kutta_ions.ipynb is a Python code demonstrating the velocity-Verlet algorithm and Runge-Kutta fourth order (RK4) algorithm for a single trapped ion in 1D. The motion equation is a second-order ODE ( $\ddot{x} = f(t,x,\dot{x})$ ). This notebook compares the dynamics obtained with velocity-Verlet and the dynamics obtained Runge-Kutta at fourth order. Runge-Kutta is implemented in two different ways. This code also read data output from the last Fortran code for another comparison. Created with jupyter notebook and Python 3.7.

Runge_Kutta_1ion.F90 is a Fortran code demonstrating the Runge-Kutta at fourth order algorithm with the same equation of motion of a single ion in 1D as in Runge_Kutta_ions.ipynb. The data from the dynamics is generated with this code for later analysis.

Open Runge_Kutta_ions.ipynb to see comparison of all methods with graphs. Those programs can be compiled with gfortran.

3D Codes

Runge_Kutta_Nions.F90 is a functional program designed to render the molecular dynamics of a collection of $N>1$ charged particles in a radio-frequency and static field, with Coulomb interaction and friction, using RK4 integration algorithm.

This program requires ifort compiler and relies on OpenMP for parallelization.

Licence

Licence Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0).