dorsal/arxiv
View SchemaStudy Notes on Numerical Solutions of the Wave Equation with the Finite Difference Method
| Authors | Artur B. Adib |
|---|---|
| Categories | |
| ArXiv ID | physics/0009068 |
| URL | https://arxiv.org/abs/physics/0009068 |
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 so-called 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.
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"abstract": "In this introductory work I will present the Finite Difference method for\nhyperbolic equations, focusing on a method which has second order precision\nboth in time and space (the so-called staggered leapfrog method) and applying\nit to the case of the 1d and 2d wave equation. A brief derivation of the energy\nand equation of motion of a wave is done before the numerical part in order to\nmake the transition from the continuum to the lattice clearer. To illustrate\nthe extension of the method to more complex equations, I also add dissipative\nterms of the kind $-\\eta \\dot{u}$ into the equations. The von Neumann numerical\nstability analysis and the Courant criterion, two of the most popular in the\nliterature, are briefly discussed. In the end I present some numerical results\nobtained with the leapfrog algorithm, illustrating the importance of the\nlattice resolution through energy plots.",
"arxiv_id": "physics/0009068",
"authors": [
"Artur B. Adib"
],
"categories": [
"physics.comp-ph",
"cond-mat",
"hep-lat"
],
"title": "Study Notes on Numerical Solutions of the Wave Equation with the Finite Difference Method",
"url": "https://arxiv.org/abs/physics/0009068"
},
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