We focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier-Stokes equations for a Newtonian and viscous fluid in contraction-expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

The behavior of a class of solutions of the shallow water Airy system originating from initial data with discontinuous derivatives is considered. Initial data are obtained by splicing together self-similar parabolae with a constant background state. These solutions are shown to develop velocity and surface gradient catastrophes in finite time and the inherent persistence of dry spots is shown to be terminated by the collapse of the parabolic core. All details of the evolution can be obtained in closed form until the collapse time, thanks to formation of simple waves that sandwich the evolving self-similar core. The continuation of solutions asymptotically for short times beyond the collapse is then investigated analytically, in its weak form, with an approach using stretched coordinates inspired by singular perturbation theory. This approach allows to follow the evolution after collapse by implementing a spectrally accurate numerical code, which is developed alongside a classical shock-capturing scheme for accuracy comparison. The codes are validated on special classes of initial data, in increasing order of complexity, to illustrate the evolution of the dry spot initial conditions on longer time scales past collapse.

In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

The classical dam-break problem for the shallow water system with a dry/vacuum downstream state is revisited in the context of exact solutions which generalize the Riemann setup of a Heaviside jump between constant states to continuous initial data. Two main setups are considered, chosen to illustrate how local properties of the dependent variables at the vacuum point influence the evolution over different time scales. For the first case, the elevation (density) variable is initially continuous but not differentiable at the dry (vacuum) point: no gradient catastrophe develops in this variable, with the time evolution eventually merging into a single (shifted) Stoker wave. Conversely, for the second case, the elevation joins the dry state with vanishing first derivative and a curvature jump: for an instant in time a global gradient catastrophe at the fixed contact point forms and immediately evolves as a Stoker parabolic simple wave, allowing the contact point to split into two moving points, one at the dry bed and one at a fixed elevation, where curvature singularities persist at all times. Although in both cases shocks develop for the velocity field, these are nongeneric, in that, in contrast to the usual case, infinitely many conservation laws are satisfied at all times. Long time evolution is further analyzed with the help of new stretched``unfolding"" variables to extract the details of the asymptotic approach to a rarefaction Stoker-like wave.

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.