Numerical development of topological and shock singularities in fluids

In the first part of this talk, I will describe a new method for resolving topological singularities, such as coalescence and pinching, in evolving fluid interfaces. The method uses a hybrid sharp–diffuse interface representation, with localized Cahn–Hilliard dynamics during topological events. In the second part, I will describe a numerical method for tracking the formation and subsequent development of one-dimensional shocks, together with the associated weak characteristic discontinuities and cusp structures. This method uses a moving mesh adapted to the evolving characteristic wave families. In both settings, I will demonstrate orders-of-magnitude gains in accuracy and efficiency over conventional numerical schemes.