SHM

Finite Difference Simple Harmonic Motion Solver

This repo contains files for solving a simple harmonic oscillator [with $k=1$, $m=1$] using two Finite Difference inspired techniques:

1. A 'forward solver' The initial conditions $[x(0) = 1, v(0) = 0]$ are converted to 'boundary' conditions $[x(0) = 1, x(1) = 1]$ through applying the discrete first derivative [difference equation: $x(n+h) -x(n) / h$ ] to the initial conditions. A central finite difference equation $[x(t+h) = (2-h^2)*x(t) -x(t-h)]$ is then applied to each grid point sequentially
2. A matrix solution, implementing the central finite difference method based on the more standard method of solving FD problems The boundary conditions are the same as approach 1) above

Files:
Python functions are saved in: shm.py
A script comparing the two approaches described above with the analytical solution is found at: shm_script.py
Performance at different grid resolutions is analyzed in: shm_perf.py

Comparison with C

The 'forward solver' is implemented in C (shm.c) and compiled in a shared library (shm.so - working in Debian 13 )

The C forward solver (implemented using Ctypes) is compared with the Python based forward solver in: shm_dt_analysis.py

shm_time_analysis.png contained in /figures shows the computation time improvements between the techniques as the number of grid points is increased