arXiv subjects

Elena Gaburro

Publications and source records attributed to Elena Gaburro.

25 records · Page 2Linked to original sources

arXiv2020

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer-Nunziato model

In this paper we propose a new diffuse interface model for the numerical simulation of inviscid compressible flows around fixed and moving solid bodies of arbitrary shape. The solids are assumed to be moving rigid bodies, without any elastic properties. The model is a simplified case of the seven-equation Baer-Nunziato model of compressible multi-phase flows, and results in a nonlinear hyperbolic system with non-conservative products. The geometry of the solid bodies is simply specified via a scalar field that represents the volume fraction of the fluid present in each control volume. This allows the discretization of arbitrarily complex geometries on simple uniform or adaptive Cartesian meshes. Inside the solid bodies, the fluid volume fraction is zero, while it is unitary in the fluid phase. We prove that at the material interface, i.e. where the volume fraction jumps from unity to zero, the normal component of the fluid velocity assumes the value of the normal component of the solid velocity. This result can be directly derived from the governing equations, either via Riemann invariants or from the generalized Rankine Hugoniot conditions according to the theory of Dal Maso, Le Floch and Murat, which justifies the use of a path-conservative approach for treating the nonconservative products. The governing equations of our new model are solved on uniform Cartesian grids via a high order path-conservative ADER discontinuous Galerkin (DG) method with a posteriori sub-cell finite volume (FV) limiter. Since the numerical method is of the shock capturing type, the fluid-solid boundary is never explicitly tracked by the numerical method, neither via interface reconstruction, nor via mesh motion. The effectiveness of the proposed approach is tested on a set of numerical test problems, including 1D Riemann problems as well as supersonic flows over fixed and moving rigid bodies.

math.NA↗

arXiv2020

High order ADER schemes for continuum mechanics

In this paper we first review the development of high order ADER finite volume and ADER discontinuous Galerkin schemes on fixed and moving meshes, since their introduction in 1999 by Toro et al. We show the modern variant of ADER based on a space-time predictor-corrector formulation in the context of ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter on fixed and moving grids, as well as on space-time adaptive Cartesian AMR meshes. We then present and discuss the unified symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of continuum mechanics developed by Godunov, Peshkov and Romenski (GPR model), which allows to describe fluid and solid mechanics in one single and unified first order hyperbolic system. In order to deal with free surface and moving boundary problems, a simple diffuse interface approach is employed, which is compatible with Eulerian schemes on fixed grids as well as direct Arbitrary-Lagrangian-Eulerian methods on moving meshes. We show some examples of moving boundary problems in fluid and solid mechanics.

physics.comp-ph↗

arXiv2019

On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations

In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of Munz et al. (JCP 2000, JCP 2002), which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type involutions, to the case of hyperbolic PDE systems with curl-type involutions. The key idea here is to solve an augmented PDE system, in which curl errors propagate away via a Maxwell-type evolution system. The new approach is first presented on a simple model problem, in order to explain the basic ideas. Subsequently, we apply it to a strongly hyperbolic first order reduction of the CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, which is endowed with 11 curl constraints. Several numerical examples, including the long-time evolution of a stable neutron star in anti-Cowling approximation, are presented in order to show the obtained improvements with respect to the standard formulation without special treatment of the curl involution constraints. The main advantages of the proposed GLM approach are its complete independence of the underlying numerical scheme and grid topology and its easy implementation into existing computer codes. However, this flexibility comes at the price of needing to add for each curl involution one additional 3 vector plus another scalar in the augmented system for homogeneous curl constraints, and even two additional scalars for non-homogeneous curl involutions. For the FO-CCZ4 system with 11 homogeneous curl involutions, this means that additional 44 evolution quantities need to be added.

math.NA↗

arXiv2019

High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes

We present a new family of very high order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on moving 2D Voronoi meshes that are regenerated at each time step and which explicitly allow topology changes in time. The Voronoi tessellations are obtained from a set of generator points that move with the local fluid velocity. We employ an AREPO-type approach, which rapidly rebuilds a new high quality mesh rearranging the element shapes and neighbors in order to guarantee a robust mesh evolution even for vortex flows and very long simulation times. The old and new Voronoi elements associated to the same generator are connected to construct closed space--time control volumes, whose bottom and top faces may be polygons with a different number of sides. We also incorporate degenerate space--time sliver elements, needed to fill the space--time holes that arise because of topology changes. The final ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and space--time sliver elements. Our new numerical scheme is based on the integration over arbitrary shaped closed space--time control volumes combined with a fully-discrete space--time conservation formulation of the governing PDE system. In this way the discrete solution is conservative and satisfies the GCL by construction. Numerical convergence studies as well as a large set of benchmarks for hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and robustness of the proposed method. Our numerical results clearly show that the new combination of very high order schemes with regenerated meshes with topology changes lead to substantial improvements compared to direct ALE methods on conforming meshes.

math.NA↗

arXiv2018

A well balanced diffuse interface method for complex nonhydrostatic free surface flows

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.

math.NA↗

arXiv2018

Well balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gasdynamics with gravity

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.

math.NA↗

arXiv2017

Direct Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes

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.

math.NA↗