arXiv subjects

Elena Gaburro

Publications and source records attributed to Elena Gaburro.

At least 19 recordsLinked to original sources

arXiv2026

On the treatment of topology changes on 3D polyhedral moving meshes via 4D space-time hole-like elements in direct ALE ADER-DG methods

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.

math.NA↗

arXiv2026

A structure-preserving semi-implicit finite volume scheme on vertex-staggered unstructured meshes

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.

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arXiv2026

Stability analysis of Arbitrary-Lagrangian-Eulerian ADER-DG methods on classical and degenerate spacetime geometries

In this paper, we present a thorough von Neumann stability analysis of explicit and implicit Arbitrary-Lagrangian-Eulerian (ALE) ADER discontinuous Galerkin (DG) methods on classical and degenerate spacetime geometries for hyperbolic equations. First, we rigorously study CFL stability conditions for the explicit ADER-DG method, confirming results widely used in the literature while specifying their limitations. Moreover, we discuss stability bounds for ALE methods and characterize the admissible range of grid velocities once a target CFL is fixed. Next, we extend the stability study to ADER-DG in the presence of degenerate spacetime elements, with zero size at the beginning and the end of the time step, but with a non zero spacetime volume. This kind of elements has been introduced in a series of articles on direct ALE methods by Gaburro et al. to connect via spacetime control volumes regenerated Voronoi tessellations after a topology change. Here, we imitate this behavior in a 1d surrogate setting by fictitiously inserting degenerate elements in between two cells. We show that over this simplified degenerate spacetime geometry, both for the explicit and implicit ADER-DG, the von Neumann analysis leads to the same CFL stability conditions as those for classical geometries, laying the theoretical foundations for their use in the context of ALE methods.

math.NA↗

arXiv2025

High order numerical discretizations of the Einstein-Euler equations in the Generalized Harmonic formulation

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.

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arXiv2025

On general and complete multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws

In this work, we introduce a framework to design multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws on general unstructured polygonal Voronoi-like tessellations. In this framework we propose two simple but complete solvers. The first method is a direct extension of the Osher-Solomon Riemann solver to multiple space dimensions. Here, the multidimensional numerical dissipation is obtained by integrating the absolute value of the flux Jacobians over a dual triangular mesh around each node of the primal polygonal grid. The required nodal gradient is then evaluated on a local computational simplex involving the d+1 neighbors meeting at each corner. The second method is a genuinely multidimensional upwind flux. By introducing a fluctuation form of finite volume methods with corner fluxes, we show an equivalence with residual distribution schemes (RD). This naturally allows to construct genuinely multidimensional upwind corner fluxes starting from existing and well-known RD fluctuations. In particular, we explore the use of corner fluxes obtained from the so-called N scheme, i.e. the Roe's original optimal multidimensional upwind advection scheme. Both methods use the full eigenstructure of the underlying hyperbolic system and are therefore complete by construction. A simple higher order extension up to fourth order in space and time is then introduced at the aid of a CWENO reconstruction in space and an ADER approach in time, leading to a family of high order accurate fully-discrete one-step schemes based on genuinely multidimensional Riemann solvers. We present applications of our new numerical schemes to several test problems for the compressible Euler equations of gas-dyanamics. In particular, we show that the proposed schemes are at the same time carbuncle-free and preserve certain stationary shear waves exactly.

math.NA↗

arXiv2024

Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: quasi-conservative formulation with subcell finite volume corrections

We present a novel quasi-conservative arbitrary high order accurate ADER discontinuous Galerkin (DG) method allowing to efficiently use a non-conservative form of the considered partial differential system, so that the governing equations can be solved directly in the most physically relevant set of variables. This is particularly interesting for multi-material flows with moving interfaces and steep, large magnitude contact discontinuities, as well as in presence of highly non-linear thermodynamics. However, the non-conservative formulation of course introduces a conservation error which would normally lead to a wrong approximation of shock waves. Hence, from the theoretical point of view, we give a formal definition of the conservation defect of non-conservative schemes and we analyze this defect providing a local quasi-conservation condition, which allows us to prove a modified Lax-Wendroff theorem. Then, to deal with shock waves in practice, we exploit the framework of the so-called a posteriori subcell finite volume (FV) limiter, so that, in troubled cells appropriately detected, we can incorporate a local conservation correction. Our corrected FV update entirely removes the local conservation defect, allowing, at least formally, to fit in the hypotheses of the proposed modified Lax-Wendroff theorem. Here, the shock-triggered troubled cells are detected by combining physical admissibility criteria, a discrete maximum principle and a shock sensor inspired by Lagrangian hydrodynamics. To prove the capabilities of our novel approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude by showing the improved reliability of our scheme on the multi-fluid Euler system on examples like the interaction of a shock with a helium bubble.

math.NA↗

arXiv2024

High order well-balanced Arbitrary-Lagrangian-Eulerian ADER discontinuous Galerkin schemes on general polygonal moving meshes

In this work, we present a novel family of high order accurate numerical schemes for the solution of hyperbolic partial differential equations (PDEs) which combines several geometrical and physical structure preserving properties. First, we settle our methods in the Lagrangian framework, where each element of the mesh evolves following as close as possible the local fluid flow, so to reduce the numerical dissipation at contact waves and moving interfaces and to satisfy the Galilean and rotational invariance properties of the studied PDEs system. In particular, we choose the direct Arbitrary-Lagrangian-Eulerian (ALE) approach which, in order to always guarantee the high quality of the moving mesh, allows to combine the Lagrangian motion with mesh optimization techniques. The employed polygonal tessellation is thus regenerated at each time step, the previous one is connected with the new one by space-time control volumes, including hole-like sliver elements in correspondence of topology changes, over which we integrate a space-time divergence form of the original PDEs through a high order accurate ADER discontinuous Galerkin (DG) scheme. Mass conservation and adherence to the GCL condition are guaranteed by construction thanks to the integration over closed control volumes, and robustness over shock discontinuities is ensured by the use of an a posteriori subcell finite volume (FV) limiting technique.

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arXiv2024

A well-balanced discontinuous Galerkin method for the first--order Z4 formulation of the Einstein--Euler system

In this paper we develop a new well-balanced discontinuous Galerkin (DG) finite element scheme with subcell finite volume (FV) limiter for the numerical solution of the Einstein--Euler equations of general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Z4 system, which is composed of 59 equations, is analyzed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows. Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching three achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum ($ρ$=$p$=0) up to t=1000 with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered un--tractable within the Z4 formalism.

gr-qc↗

arXiv2023

Very high order treatment of embedded curved boundaries in compressible flows: ADER discontinuous Galerkin with a space-time Reconstruction for Off-site data

In this paper we present a novel approach for the design of high order general boundary conditions when approximating solutions of the Euler equations on domains with curved boundaries, using meshes which may not be boundary conformal. When dealing with curved boundaries and/or unfitted discretizations, the consistency of boundary conditions is a well-known challenge, especially in the context of high order schemes. In order to tackle such consistency problems, the so-called Reconstruction for Off-site Data (ROD) method has been recently introduced in the finite volume framework: it is based on performing a boundary polynomial reconstruction that embeds the considered boundary treatment thanks to the implementation of a constrained minimization problem. This work is devoted to the development of the ROD approach in the context of discontinuous finite elements. We use the genuine space-time nature of the local ADER predictors to reformulate the ROD as a single space-time reconstruction procedure. This allows us to avoid a new reconstruction (linear system inversion) at each sub-time node and retrieve a single space-time polynomial that embeds the considered boundary conditions for the entire space-time element. Several numerical experiments are presented proving the consistency of the new approach for all kinds of boundary conditions. Computations involving the interaction of shocks with embedded curved boundaries are made possible through an a posteriori limiting technique.

math.NA↗

arXiv2023

Second Order Finite Volume Scheme for Shallow Water Equations on Manifolds

In this work, we propose a second-order accurate scheme for shallow water equations in general covariant coordinates over manifolds. In particular, the covariant parametrization in general covariant coordinates is induced by the metric tensor associated to the manifold. The model is then re-written in a hyperbolic form with a tuple of conserved variables composed both of the evolving physical quantities and the metric coefficients. This formulation allows the numerical scheme to i) automatically compute the curvature of the manifold as long as the physical variables are evolved and ii) numerically study complex physical domains over simple computational domains.

math.NA↗

arXiv2022

Well balanced finite volume schemes for shallow water equations on manifolds

In this paper we propose a novel second-order accurate well balanced scheme for shallow water equations in general covariant coordinates over manifolds. In our approach, once the gravitational field is defined for the specific case, one equipotential surface is detected and parametrized by a frame of general covariant coordinates. This surface is the manifold whose covariant parametrization induces a metric tensor. The model is then re-written in a hyperbolic form with a tuple of conserved variables composed both of the evolving physical quantities and the metric coefficients. This formulation allows the numerical scheme to automatically compute the curvature of the manifold as long as the physical variables are evolved.

math.NA↗

arXiv2022

Shifted boundary polynomial corrections for compressible flows: high order on curved domains using linear meshes

In this work we propose a simple but effective high order polynomial correction allowing to enhance the consistency of all kind of boundary conditions for the Euler equations (Dirichlet, characteristic far-field and slip-wall), both in 2D and 3D, preserving a high order of accuracy without the need of curved meshes. The method proposed is a simplified reformulation of the Shifted Boundary Method (SBM) and relies on a correction based on the extrapolated value of the in cell polynomial to the true geometry, thus not requiring the explicit evaluation of high order Taylor series. Moreover, this strategy could be easily implemented into any already existing finite element and finite volume code. Several validation tests are presented to prove the convergence properties up to order four for 2D and 3D simulations with curved boundaries, as well as an effective extension to flows with shocks.

math.NA↗

arXiv2022

High-order Arbitrary-Lagrangian-Eulerian schemes on crazy moving Voronoi meshes

Hyperbolic partial differential equations (PDEs) cover a wide range of interesting phenomena, from human and hearth-sciences up to astrophysics: this unavoidably requires the treatment of many space and time scales in order to describe at the same time observer-size macrostructures, multi-scale turbulent features, and also zero-scale shocks. Moreover, numerical methods for solving hyperbolic PDEs must reliably handle different families of waves: smooth rarefactions, and discontinuities of shock and contact type. In order to achieve these goals, an effective approach consists in the combination of space-time-based high-order schemes, very accurate on smooth features even on coarse grids, with Lagrangian methods, which, by moving the mesh with the fluid flow, yield highly resolved and minimally dissipative results on both shocks and contacts. However, ensuring the high quality of moving meshes is a huge challenge that needs the development of innovative and unconventional techniques. The scheme proposed here falls into the family of Arbitrary-Lagrangian-Eulerian (ALE) methods, with the unique additional freedom of evolving the shape of the mesh elements through connectivity changes. We aim here at showing, by simple and very salient examples, the capabilities of high-order ALE schemes, and of our novel technique, based on the high-order space-time treatment of topology changes.

math.NA↗

arXiv2022

High order entropy preserving ADER scheme

In this paper, we develop a fully discrete entropy preserving ADER-Discontinuous Galerkin (ADER-DG) method. To obtain this desired result, we equip the space part of the method with entropy correction terms that balance the entropy production in space, inspired by the work of Abgrall. Whereas for the time-discretization we apply the relaxation approach introduced by Ketcheson that allows to modify the timestep to preserve the entropy to machine precision. Up to our knowledge, it is the first time that a provable fully discrete entropy preserving ADER-DG scheme is constructed. We verify our theoretical results with various numerical simulations.

math.NA↗

arXiv2022

Continuous finite element subgrid basis functions for Discontinuous Galerkin schemes on unstructured polygonal Voronoi meshes

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree N inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree N within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor. The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.

math.NA↗

arXiv2021

A well balanced finite volume scheme for general relativity

In this work we present a novel second order accurate well balanced (WB) finite volume (FV) scheme for the solution of the general relativistic magnetohydrodynamics (GRMHD) equations and the first order CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, as well as the fully coupled FO-CCZ4 + GRMHD system. These systems of first order hyperbolic PDEs allow to study the dynamics of the matter and the dynamics of the space-time according to the theory of general relativity. The new well balanced finite volume scheme presented here exploits the knowledge of an equilibrium solution of interest when integrating the conservative fluxes, the nonconservative products and the algebraic source terms, and also when performing the piecewise linear data reconstruction. This results in a rather simple modification of the underlying second order FV scheme, which, however, being able to cancel numerical errors committed with respect to the equilibrium component of the numerical solution, substantially improves the accuracy and long-time stability of the numerical scheme when simulating small perturbations of stationary equilibria. In particular, the need for well balanced techniques appears to be more and more crucial as the applications increase their complexity. We close the paper with a series of numerical tests of increasing difficulty, where we study the evolution of small perturbations of accretion problems and stable TOV neutron stars. Our results show that the well balancing significantly improves the long-time stability of the finite volume scheme compared to a standard one.

gr-qc↗

arXiv2021

A simple but efficient concept of blended teaching of mathematics for engineering students during the COVID-19 pandemic

We present a simple but efficient concept for the realization of blended teaching of mathematics and its applications in theoretical mechanics that was conceived, tested and implemented at the University of Trento, Italy, during the COVID-19 pandemic. The concept foresees traditional blackboard lectures with a reduced number of students present in the lecture hall, while the same lectures are simultaneously made available to the remaining students via high quality low-bandwidth online streaming. Based on our first assumption that traditional blackboard lectures, including the gestures and the facial expressions of the professor, are still a very efficient and highly appreciated means of teaching mathematics, this paper deliberately does not want to propose a novel pedagogical concept of how to teach mathematics, but rather presents a technical concept how to preserve the quality of traditional blackboard lectures even during the pandemic and how to make them available to the students at home via online streaming with adequate audio and video quality at low internet bandwidth. The second assumption is that the teaching of mathematics is a dynamic creative process that requires the physical presence of students in the lecture hall as audience so that the professor can instantaneously fine-tune the evolution of the lecture according to his/her perception of the level of attention and the facial expressions of the students. The third assumption of this paper is that students need to have the possibility to interact with each other personally. We report on the necessary hardware, software and logistics, and on the perception of the proposed blended lectures by students from civil and environmental engineering at the University of Trento, compared to traditional lectures and also compared to the pure online lectures that were needed as emergency measure at the beginning of the pandemic.

math.HO↗

arXiv2020

A posteriori subcell finite volume limiter for general PNPM schemes: applications from gasdynamics to relativistic magnetohydrodynamics

In this work, we consider the general family of the so called ADER PNPM schemes for the numerical solution of hyperbolic partial differential equations with \textit{arbitrary} high order of accuracy in space and time. The family of one-step PNPM schemes was introduced in [Dumbser et al., JCP, 2008] and represents a unified framework for classical high order Finite Volume (FV) schemes (N=0), the usual Discontinuous Galerkin (DG) methods (N=M), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with M>N. In all cases with M >= N > 0 the PNPM schemes are linear in the sense of Godunov, thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of PNPM schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order PNPM schemes, due to the use of a rather fine subgrid of 2N+1 subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes (AMR) that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

math.NA↗