Numerical computation of wave breaking times

The time of nonlinear wave collapse is computed numerically for the Hopf equation. Previous numerical criteria for locating the time of wave collapse are either computationally prohibitive to implement or give erroneous results. A new criterion for this purpose is developed analytically using asymptotic analysis of the wave shock development shortly after breaking. The criterion defines the wave breaking time as the onset of energy dissipation. This onset results in a singularity in the third derivative of the energy function, the location of which yields the breaking time. A numerical criterion is formulated from this analytical result and tested against the exact analytical value of the breaking time. This is done by first solving the differential equation with a finite difference method. Then, to compensate for numerical error, a moving average method is developed to refine the energy data. The obtained results give visible convergence to the analytical breaking time as the numerical mesh is refined.