Local Formulations of Nonlocal Problems: Dispersive Tsunamis, Nonlinear Schrödinger Systems, and Absorbing Boundaries

We analyze the behavior of several hyperbolic approximations of the Serre-Green-Naghdi Equations, and develop an adaptive and efficient solver for dispersive tsunami simulation within the GeoClaw software. The performance of the solver and the practical potential of these hyperbolic models are validated against theoretical and experimental tests as well as realistic scenarios, yielding promising results. Then, we consider a novel hyperbolized Nonlinear Schrödinger System. We study in depth several theoretical properties of the new model, such as its conservative structure, dispersion relation, and stability of plane wave solutions. Several analytical special solutions are proposed, including standing and traveling waves. Regimes in which the model behaves qualitatively similar to the Nonlinear Schrödinger Equation, as well as those in which they significantly differ, are analyzed. Furthermore, we propose a variety of numerical techniques for its treatment, including adaptive finite volume methods for problems with challenging superfluid flows. Finally, we study the imposition of absorbing boundary conditions for one-dimensional conservation laws, with emphasis on the piston problem of gas dynamics, described by the polytropic Euler equations in Lagrangian form.