The Small Dispersion Limit for a Nonlinear Semidiscrete System of Equations

Abstract

Numerical experiments that illustrate the emergence of oscillatory behavior in the solutions of a dispersive scheme approximating the Hopf equation are presented. The oscillations arise at the same time that the classical solution of the Hopf equation develops a singularity and generally have a spatial period equal to twice the grid size. Modulation equations for these period‐two solutions are derived. The modulation equations have both a hyperbolic and an elliptic region. The period‐two oscillations break down after they enter the elliptic region, and the solution blows up. We give a local description of the blowup by an exact solution. This kind of phenomenon (the blowup) has not been observed for integrable schemes. The modulation equations also have the unusual feature that they admit (some) shocks when crossings of characteristics in the hyperbolic regime occur. Other crossings lead to breakdown of the binary oscillation description, with oscillatory behavior of a more complicated nature arising.