Study Notes on Numerical Solutions of the Wave Equation with the Finite Difference Method
Abstract
In this introductory work I will present the Finite Difference method for hyperbolic equations, focusing on a method which has second order precision both in time and space (the socalled staggered leapfrog method) and applying it to the case of the 1d and 2d wave equation. A brief derivation of the energy and equation of motion of a wave is done before the numerical part in order to make the transition from the continuum to the lattice clearer. To illustrate the extension of the method to more complex equations, I also add dissipative terms of the kind $\eta \dot{u}$ into the equations. The von Neumann numerical stability analysis and the Courant criterion, two of the most popular in the literature, are briefly discussed. In the end I present some numerical results obtained with the leapfrog algorithm, illustrating the importance of the lattice resolution through energy plots.
 Publication:

arXiv eprints
 Pub Date:
 September 2000
 arXiv:
 arXiv:physics/0009068
 Bibcode:
 2000physics...9068A
 Keywords:

 Physics  Computational Physics;
 Condensed Matter;
 High Energy Physics  Lattice
 EPrint:
 20 pages, 7 figures. Study supported by the PIBIC/CNPq undergraduate research program, Brazil. Leapfrog section completely rewritten and some corrected typos