Levermore, CD; Liu, J-G, *Large oscillations arising in a dispersive numerical scheme*,
Physica D: Nonlinear Phenomena, vol. 99 no. 2-3
(1996),
pp. 191-216 .
**Abstract:**

*We study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the Hopf equation whenever the classical solution of that equation develops a singularity. Modulation equations are derived that describe period-two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic. However, those equations have an elliptic region that may be entered by its solutions in a unite time, after which the corresponding period-two oscillations are seen to break down. This kind of phenomenon has not been observed for integrable schemes. The generation and propagation of period-two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions. Modulation equations are also presented for period-three oscillations. Numerical experiments are carried out that illustrate our analysis. © 1996 Elsevier Science B.V. All rights reserved.*