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.

%B Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences %V 474 %P 20170458 %G eng %U https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0458 %R 10.1098/rspa.2017.0458 %0 Journal Article %J COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING %D 2018 %T NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces %A Giuseppe Pitton %A Luca Heltai %B COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING %V 338 %P 440–462 %G eng %U https://arxiv.org/abs/1804.08271 %R 10.1016/j.cma.2018.04.039 %0 Journal Article %J Journal of Scientific Computing %D 2017 %T On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics %A Giuseppe Pitton %A Gianluigi Rozza %XIn 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.

%B Journal of Scientific Computing %G eng %R 10.1007/s10915-017-0419-6 %0 Journal Article %J Journal of Computational Physics %D 2017 %T Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology %A Giuseppe Pitton %A Annalisa Quaini %A Gianluigi Rozza %K Parametrized Navier–Stokes equations %K Reduced basis method %K Stability of flows %K Symmetry breaking bifurcation %XWe 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.