Discrete Analog of the Burgers Equation

TL;DR

This paper introduces a discrete set of differential equations as an analog to the Burgers equation, analyzing traveling and triangular wave solutions to compare their behaviors with the continuous case.

Contribution

It proposes a novel discrete analog of the Burgers equation and analyzes its wave solutions, revealing both similarities and subtle differences from the continuous model.

Findings

Traveling wave velocity matches continuous behavior

Triangular wave shape and propagation are similar to continuous case

Discovered sharp fronts and logarithmic shifts in solutions

Abstract

We propose the set of coupled ordinary differential equations dn_j/dt=(n_{j-1})^2-(n_j)^2 as a discrete analog of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wave match the continuous behavior. However, there are some subtle differences. For traveling waves, the propagating front can be extremely sharp as it exhibits double exponential decay. For triangular waves, there is an unexpected logarithmic shift in the location of the front. We establish these results using asymptotic analysis, heuristic arguments, and direct numerical integration.