Abstract

This PhD thesis presents a novel second order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume scheme for nonlinear hyperbolic systems, written both in conservative and non-conservative form, whose peculiarity is the nonconforming motion of interfaces. Moreover it has been coupled together with specifically designed path-conservative well balanced (WB) techniques and angular momentum preserving (AMC) strategies. The obtained result is a method able to preserve many of the physical properties of the system: besides being conservative for mass, momentum and total energy, also any known steady equilibrium of the studied system can be exactly maintained up to machine precision. Perturbations around such equilibrium solutions are resolved with high accuracy and minimal dissipation on moving contact discontinuities even for very long computational times. The core of our ALE scheme is the use of a space-time conservation formulation in the construction of the final Finite Volume scheme: the governing PDE system is rewritten at the aid of the space-time divergence operator and then a fully discrete one-step discretization is obtained by integrating over a set of closed space-time control volumes. In order to avoid the typical mesh distortion caused by shear flows in Lagrangian-type methods, we adopt a nonconforming treatment of sliding interfaces, which requires the dynamical insertion or deletion of nodes and edges, and produces hanging nodes and space-time faces shared between more than two cells. In this way, the elements on both sides of the shear wave can move with a different velocity, without producing highly distorted elements, the mesh quality remains high and, as a direct consequence, also the time step remains almost constant in time, even for highly sheared vortex flows. Moreover, due to the space-time conservation formulation, the geometric conservation law (GCL) is automatically satisfied by construction, even on moving nonconforming meshes. Our nonconforming ALE scheme is especially well suited for modeling in polar coordinates vortical flows affected by strong differential rotation: in particular, the novel combination with the well balancing make it possible to obtain great results for challenging astronomical phenomena as the rotating Keplerian disk. Indeed, we have formulated a new HLL-type and a novel Osher-type flux that are both able to guarantee the well balancing in a gas cloud rotating around a central object, maintaining up to machine precision the equilibrium between pressure gradient, centrifugal force and gravity force that characterizes the stationary solutions of the Euler equations with gravity. To the best knowledge of the author this work is original for various reasons: it is the first time that the little dissipative Osher scheme is modified in order to be well balanced for non trivial equilibria, and it is the first time that WB is coupled with ALE for the Euler equations with gravity; moreover the use of a well balanced Osher scheme joint with the Lagrangian framework allows, for the first time within a Finite Volume method, to maintain exactly even moving equilibria. In addition, the introduced techniques demonstrate a wide range of applicability from steady vortex flows in shallow water equations to complex free surface flows in two-phase models. In the last case, studied on fixed Cartesian grids, the new well balanced methods have been implemented in parallel exploiting a GPU-based platform and reaching the very high efficiency of ten million of volumes processed per seconds. Finally, in the case of vortical flows we propose a preliminary analysis on how to increase the accuracy of the method by exploiting the redundant conservation law that can be written for the angular momentum, as proposed in Després et al. JCP 2015. Indeed, an easy manipulation of the Euler equations allows to write its additional conservation law: clearly it does not add any supplementary information from the analytical point of view, but from a numerical point of view it provides extra information in particular in the case of rotating systems. We present both a master-slave approach, to deduce a posteriori a more precise approximation of the velocity, and some coupled approaches to investigate how the entire process can take advantage from considering directly the angular momentum during the computation within a strong coupling with other variables. A large set of different numerical tests has been carried out in order to check the accuracy and the robustness of the new methods for both smooth and discontinuous problems, close and far away from the equilibrium, in one and two space dimensions. Many of the presented results show a great enhancement with respect to the state of the art.