Elena Gaburro

Elena Gaburro, Manuel J. Castro, Michael Dumbser

In this work we present a novel second order accurate well balanced Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving nonconforming meshes for the Euler equations of compressible gasdynamics with gravity in cylindrical coordinates. The main feature of the proposed algorithm is the capability of preserving many of the physical properties of the system exactly also on the discrete level: besides being conservative for mass, momentum and total energy, also any known steady equilibrium between pressure gradient, centrifugal force and gravity force can be exactly maintained up to machine precision. Perturbations around such equilibrium solutions are resolved with high accuracy and with minimal dissipation on moving contact discontinuities even for very long computational times. This is achieved by the novel combination of well balanced path-conservative finite volume schemes, that are expressly designed to deal with source terms written via nonconservative products, with ALE schemes on moving grids, which exhibit only very little numerical dissipation on moving contact waves. In particular, 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. Moreover, to maintain a high level of quality of the moving mesh, we have adopted a nonconforming treatment of the sliding interfaces that appear due to the differential rotation. A large set of numerical tests has been carried out in order to check the accuracy of the method close and far away from the equilibrium, both, in one and two space dimensions.

Elena Gaburro, Michael Dumbser, Manuel J. Castro

In this paper, we present a novel second-order accurate Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving nonconforming polygonal grids, in order to avoid the typical mesh distortion caused by shear flows in Lagrangian-type methods. In our new approach the nonconforming element interfaces are not defined by the user, but they are automatically detected by the algorithm if the tangential velocity difference across an element interface is sufficiently large. The grid nodes that are sufficiently far away from a shear wave are moved with a standard node solver, while at the interface we insert a new set of nodes that can slide in a nonconforming manner. In this way, the elements on both sides of the shear wave can move with a different velocity, without producing highly distorted elements. The core of the proposed method is the use of a space-time conservation formulation in the construction of the final finite volume scheme, which completely avoids the need of an additional remapping stage and ensures that the geometric conservation law (GCL) is automatically satisfied. Moreover, 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. The accuracy of the new scheme has been further improved by incorporating a special well balancing technique that is able to maintain particular stationary solutions of the governing PDE system up to machine precision. A large set of numerical tests has been carried out in order to check the accuracy and the robustness of the new method. In particular we have compared the results for a steady vortex in equilibrium solved with a standard conforming ALE method and with our new nonconforming ALE scheme, to show that the new nonconforming scheme is able to avoid mesh distortion even after very long simulation times.

Elena Gaburro, Manuel J. Castro, Michael Dumbser

In this paper we propose an efficient second order well balanced finite volume method for modeling complex free surface flows at the aid of a simple diffuse interface method. The employed physical model is a two-phase model derived from the Baer-Nunziato system for compressible multi-phase flows. In particular, as proposed for the first time in Dumbser (2011), the number of equations is reduced from seven to three by assuming that the relative pressure of the gas with respect to the atmospheric reference pressure is zero, and that the gas momentum is negligible compared to the one of the liquid. The two-phase model does not make any of the classical assumptions of shallow water type systems, hence it does not neglect vertical accelerations and the free surface is not constraint to be a single-valued function, so even complex shapes as those of breaking waves can be properly captured. The resulting PDE system is solved by a novel well balanced path-conservative finite volume method on structured Cartesian grids, which is able to preserve exactly the equilibrium states even in the presence of obstacles. It furthermore automatically computes the location of the water-air interfaces, and assures low numerical dissipation at the free surface thanks to a novel Osher-Romberg-type Riemann solver. Finally, high computational performance is guaranteed by an efficient parallel implementation on a GPU-based platform that reaches the efficiency of twenty million of volumes processed per seconds and makes it possible to employ even very fine meshes. The validation of our new well balanced scheme is carried out by comparing the obtained numerical results against existing analytical, numerical and experimental reference solutions for a large number of test cases, among which oscillating elliptical drops, dambreak problems, breaking waves, over topping weir flows, and wave impact problems.

Stefano Muzzolon, Michael Dumbser, Olindo Zanotti et al.

We propose two new alternative numerical schemes to solve the coupled Einstein-Euler equations in the Generalized Harmonic formulation. The first one is a finite difference (FD) Central Weighted Essentially Non-Oscillatory (CWENO) scheme on a traditional Cartesian mesh, while the second one is an ADER (Arbitrary high order Derivatives) discontinuous Galerkin (DG) scheme on 2D unstructured polygonal meshes. The latter, in particular, represents a preliminary step in view of a full 3D numerical relativity calculation on moving meshes. Both schemes are equipped with a well-balancing (WB) property, which allows to preserve the equilibrium of a priori known stationary solutions exactly at the discrete level. We validate our numerical approaches by successfully reproducing standard vacuum test cases, such as the robust stability, the linearized wave, and the gauge wave tests, as well as achieving long-term stable evolutions of stationary black holes, including Kerr black holes with extreme spin. Concerning the coupling with matter, modeled by the relativistic Euler equations, we perform some special relativistic Riemann problems, a classical test of spherical accretion onto a Schwarzschild black hole, as well as an evolution of a perturbed non-rotating neutron star, demonstrating the capability of our schemes to operate also on the full Einstein-Euler system. Altogether, these results provide a solid foundation for addressing more complex and challenging simulations of astrophysical sources through DG schemes on unstructured 3D meshes.

Elena Gaburro, Matej Klima, Mauro Bonafini et al.

This work investigates a novel approach for the high order evolution of hyperbolic PDEs using ADER discontinuous Galerkin schemes within a direct Arbitrary-Lagrangian-Eulerian (ALE) framework on 3D moving polyhedral meshes with topology changes. Our direct ALE method is based on the PDE integration over 4D (3D+time) space-time control volumes connecting the elements of two subsequent tessellations, so to simultaneously evolve the solution both in time and between the two different meshes in an effective and high order manner. In this way, we also avoid any complex and expensive projection-reconstruction techniques and any mesh intersection operation typical of indirect ALE schemes. The crucial step consists in the strategy for building space-time control volumes that also connect elements with different shapes and neighborhoods due to a change in topology. In fact, simply linking existing elements by collapsing or expanding their edges would leave a "hole" in the space-time domain. To fill it, we introduce additional degenerate elements that we call hole-like elements. These are 4D objects with zero 3D volume at both the beginning and end of the timestep, but which possess a strictly non-zero 4D space-time volume. Given the uniqueness of this space-time approach in 3D+time and the necessity of characterizing the geometry of such elements, the main objective of this paper is the formal geometrical and numerical description of the method as well as the presentation of new and intuitive visualization strategies. In particular, we provide a detailed characterization of the hole-like elements arising in correspondence to 2-3, 3-2, and 4-4 flips on the underlying Delaunay tetrahedralization. Finally, we numerically show that the method is fully conservative, satisfies the GCL and maintains the correct order of convergence even in the presence of frequent topology changes.

Elena Bernardelli, Elena Gaburro, Michael Dumbser

We present a novel structure-preserving semi-implicit finite volume method on vertex-based staggered meshes for the compatible discretization of first order systems of time-dependent partial differential equations (PDEs). The method preserves divergence-free and curl-free vector fields exactly thanks to the compatible vertex-staggered discretization of the state variables on unstructured grids that are constituted by primal Delaunay triangles and their dual polygons. For the weakly compressible Euler equations, the scheme is asymptotic preserving, yielding a consistent discretization of the incompressible limit as the Mach number goes to zero. The new scheme applies to a broad spectrum of PDEs, including the weakly compressible and incompressible Euler and Navier-Stokes equations, the incompressible magnetohydrodynamics (MHD) system, and the incompressible version of the first-order hyperbolic Godunov-Peshkov-Romenski (GPR) model for continuum mechanics. The computational domain is covered by a primal triangular mesh and a dual tessellation made of so-called star polygons. Scalar quantities (pressure, density, viscous stress) are defined at nodes, with pressure updated implicitly in a continuous finite element fashion, yielding a symmetric and positive definite pressure system. Instead, vector fields (velocity, momentum, magnetic and distortion fields) are stored at triangle barycenters and evolved explicitly using a compatible finite volume scheme. Thanks to the semi-implicit discretization, the CFL condition is independent of the sound speed, allowing simulations at low Mach numbers. The fully compatible formulation ensures exactly divergence-free velocity field in the incompressible limit, exactly divergence-free magnetic field for MHD, and exactly curl-free inverse deformation gradient in solid mechanics. The method is validated through a wide set of test cases.