Some of the most impressive singular wave fronts seen in Nature are the transbasin oceanic internal waves, which may be observed from a space shuttle, as they propagate and interact with each other. The characteristic feature of these strongly nonlinear waves is that they reconnect whenever any two of them collide transversely. The dynamics of these internal wave fronts is governed by the so-called EPDiff equation, which, in particular, coincides with the dispersionless case of the Camassa-Holm (CH) equation for shallow water in one- and two-dimensions.
Typical weak solutions of the EPDiff equation are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. The equation admits solutions that are nonlinear superpositions of traveling waves and troughs that have a discontinuity in the first derivative at their peaks and therefore are called peakons. Capturing these solutions numerically is a challenging task especially when a peakon-antipeakon interaction needs to be resolved.
In this talk, I will present a particle method for the numerical simulation and investigation of solitary wave structures of the EPDiff equation in one and two dimensions. I will show that the discretization of the EPDiff by means of the particle method preserves the basic Hamiltonian, the weak and variational structure of the original problem, and respects the conservation laws associated with symmetry under the Euclidean group. I will also present a convergence analysis of the proposed particle method in 1-D.
Finally, I will demonstrate the performance of the particle methods on a number of numerical examples in both one and two dimensions. The numerical results illustrate that the particle method has superior features and represent huge computational savings when the initial data of interest lies on a submanifold. The method can also be effectively implemented in straightforward fashion in a parallel computing environment for arbitrary initial data.