arXiv subjects

Mirco Ciallella

Publications and source records attributed to Mirco Ciallella.

16 recordsLinked to original sources

arXiv2026

Generalized high-order minimization-based polynomial corrections on unfitted spectral elements for the Poisson problem

Higher-order finite element methods are effective for solving partial differential equations. However, applying them in complex curved domains is often difficult due to challenges in creating high-quality curvilinear meshes. Unfitted, or embedded, methods provide a valid alternative by avoiding complete mesh generation and curved element integration, but their accuracy can suffer from geometric errors introduced during embedding. In this work, we explore a new family of polynomial corrections obtained by solving a local minimization problem to improve the consistency of embedded boundary methods. This approach generalizes existing techniques such as the shifted boundary method (SBM) and the reconstruction for off-site data (ROD) method. Unlike the SBM, which depends on truncated Taylor expansions, the proposed family of polynomial corrections is derived from a constrained minimization problem, similar to the ROD method. We demonstrate that this approach yields better system conditioning than the original SBM and provides a flexible framework that can be applied pointwise, without solving the full ROD linear system for each boundary element. This paper presents four formulations based on different functionals and extends the family of minimization-based polynomial corrections to handle Neumann and Robin boundary conditions, demonstrating that an elegant formulation is possible in this context. Several numerical experiments for the Poisson problem are presented to show that the generalized polynomial corrections achieve high-order accuracy across all boundary conditions.

math.NA↗

arXiv2026

Minimization-based polynomial corrections for high-order curved boundaries on fixed and moving domains: assessment on finite volume and discontinuous Galerkin schemes

In this work, we present two novel strategies to impose high-order boundary conditions on fixed and moving curved domains, approximated with piecewise affine triangulations. Achieving high-order accuracy on curved domains requires tackling both the PDE discretization error and the geometrical error simultaneously. While the former can be reduced by employing high-order numerical methods such as finite volume and discontinuous Galerkin, the latter demands either a high-order parametrization of the physical domain or a consistent approximation of the boundary conditions. Minimization-based approaches like the Reconstruction for Off-site Data (ROD) method allow one to skip the construction of high-order curvilinear meshes by defining high-order consistent boundary conditions on a computational boundary that does not match the physical one. The ROD approach mitigates the second-order geometrical error by retrieving a modified polynomial in each boundary cell, which enforces the boundary conditions exactly on the physical boundary. However, the standard ROD method requires the inversion of a local linear system, whose cost grows with the polynomial degree and mesh refinement. Inspired by a recent one-dimensional analysis, we show that ROD-type approaches can be recast as simple polynomial corrections, applicable without any linear system inversion. This greatly simplifies the development of minimization-based boundary treatments and reduces the associated computational cost. To prove the wide applicability of our strategy, we develop it within a Runge-Kutta discontinuous Galerkin framework and an ADER arbitrary-Lagrangian-Eulerian finite volume framework for compressible flows on fixed and moving domains. Several numerical experiments with Dirichlet and slip-wall boundary conditions are presented, with convergence analysis up to fifth order in both 2D and 3D.

math.NA↗

arXiv2025

Minimization-based embedded boundary methods as polynomial corrections: a stability study of discontinuous Galerkin for hyperbolic equations

This work establishes a novel, unified theoretical framework for a class of high order embedded boundary methods, revealing that the Reconstruction for Off-site Data (ROD) treatment shares a fundamental structure with the recently developed shifted boundary polynomial correction [Ciallella, M., et al. (2023)]. By proving that the ROD minimization problem admits an equivalent direct polynomial correction formulation, we unlock two major advances. First, we derive a significant algorithmic simplification, replacing the solution of the minimization problem with a straightforward polynomial evaluation, thereby enhancing computational efficiency. Second, and most critically, this reformulation enables the first stability result for the ROD method when applied to the linear advection equation with discontinuous Galerkin discretization. Our analysis, supported by a comprehensive eigenspectrum study for polynomial degrees up to six, characterizes the stability region of the new ROD formulation. The theoretical findings, which demonstrate the stability constraints, are validated through targeted numerical experiments.

math.NA↗

arXiv2025

Genuinely multi-dimensional stationarity preserving Finite Volume formulation for nonlinear hyperbolic PDEs

Classical Finite Volume methods for multi-dimensional problems include stabilization (e.g.\ via a Riemann solver), that is derived by considering several one-dimensional problems in different directions. Such methods therefore ignore a possibly existing balance of contributions coming from different directions, such as the one characterizing multi-dimensional stationary states. Instead of being preserved, they are usually diffused away by such methods. Stationarity preserving methods use a better suited stabilization term that vanishes at the stationary state, allowing the method to preserve it. This work presents a general approach to stationarity preserving Finite Volume methods for nonlinear conservation/balance laws. It is based on a multi-dimensional stationarity preserving quadrature strategy that allows to naturally introduce genuinely multi-dimensional numerical fluxes. The new methods are shown to significantly outperform existing ones even if the latter are of higher order of accuracy and even on non-stationary solutions.

math.NA↗

arXiv2025

Stability analysis of discontinuous Galerkin with a high order embedded boundary treatment for linear hyperbolic equations

Embedded, or immersed, approaches have the goal of reducing to the minimum the computational costs associated with the generation of body-fitted meshes by only employing fixed, possibly Cartesian, meshes over which complex boundaries can move freely. However, this boundary treatment introduces a geometrical error of the order of the mesh size that, if not treated properly, can spoil the global accuracy of a high order discretization, herein based on discontinuous Galerkin. The shifted boundary polynomial correction was proposed as a simplified version of the shifted boundary method, which is an embedded boundary treatment based on Taylor expansions to deal with unfitted boundaries. It is used to accordingly correct the boundary conditions imposed on a non-meshed boundary to compensate the aforementioned geometrical error, and reach high order accuracy. In this paper, the stability analysis of discontinuous Galerkin methods coupled with the shifted boundary polynomial correction is conducted in depth for the linear advection equation, by visualizing the eigenvalue spectrum of the high order discretized operators. The analysis considers a simplified one-dimensional setting by varying the degree of the polynomials and the distance between the real boundary and the closest mesh interface. The main result of the analysis shows that the considered high order embedded boundary treatment introduces a limitation to the stability region of high order discontinuous Galerkin methods with explicit time integration, which becomes more and more important when using higher order methods. The implicit time integration is also studied, showing that the implicit treatment of the boundary condition allows one to overcome such limitation and achieve an unconditionally stable high order embedded boundary treatment.

math.NA↗

arXiv2025

High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products

Shallow water moment equations are reduced-order models for free-surface flows that allow to represent vertical variations of the velocity profile at the expense of additional evolution equations for a number of additional variables, so called moments. This introduces non-linear non-conservative products in the system, which make the analytical characterization of steady states much harder if not impossible. The lack of analytical steady states poses a challenge for the design of well-balanced schemes, which aim at preserving such steady states as crucial in many applications. In this work, we present a family of fully well-balanced, high-order WENO finite volume methods for general hyperbolic balance laws with non-conservative products like the shallow water moment equations, for which no analytical steady states are available. The schemes are based on the flux globalization approach, in which both source terms and non-conservative products are integrated with a tailored high order quadrature in the divergence term. The resulting global flux is then reconstructed instead of the conservative variables to preserve all steady states. Numerical tests show the optimal convergence of the method and a significant error reduction for steady state solutions. Furthermore, we provide a numerical comparison of perturbed steady states for different families of shallow water moment equations, which illustrates the flexibility of our method that is valid for general equations without prior knowledge of steady states.

math.NA↗

arXiv2025

High order treatment of moving curved boundaries: Arbitrary-Lagrangian-Eulerian methods with a shifted boundary polynomials correction

In this paper we present a novel approach for the prescription of high order boundary conditions when approximating the solution of the Euler equations for compressible gas dynamics on curved moving domains. When dealing with curved boundaries, the consistency of boundary conditions is a real challenge, and it becomes even more challenging in the context of moving domains discretized with high order Arbitrary-Lagrangian-Eulerian (ALE) schemes. The ALE formulation is particularly well-suited for handling moving and deforming domains, thus allowing for the simulation of complex fluid-structure interaction problems. However, if not properly treated, the imposition of boundary conditions can lead to significant errors in the numerical solution, which can spoil the high order discretization of the underlying mathematical model. In order to tackle this issue, we propose a new method based on the recently developed shifted boundary polynomial correction, which was originally proposed on fixed meshes. The new method is integrated into the space-time corrector step of a direct ALE finite volume method to account for the local curvature of the moving boundary by only exploiting the high order reconstruction polynomial of the finite volume control volume. It relies on a correction based on the extrapolated value of the cell polynomial evaluated at the true geometry, thus not requiring the explicit evaluation of high order Taylor series. This greatly simplifies the treatment of moving curved boundaries, as it allows for the use of standard simplicial meshes, which are much easier to generate and move than curvilinear ones, especially for 3D time-dependent problems. Several numerical experiments are presented demonstrating the high order convergence properties of the new method in the context of compressible flows in moving curved domains, which remain approximated by piecewise linear elements.

math.NA↗

arXiv2024

Fully Eulerian models for the numerical simulation of capsules with an elastic bulk nucleus

In this paper, we present a computational framework based on fully Eulerian models for fluid-structure interaction for the numerical simulation of biological capsules. The flexibility of such models, given by the Eulerian treatment of the interface and deformations, allows us to easily deal with the large deformations experienced by the capsule. The modeling of the membrane is based on a full membrane elasticity Eulerian model that is capable of capturing both area and shear variations thanks to the so-called backward characteristics. In the validation section several test cases are presented with the goal of comparing our results to others present in the literature. In this part, the comparisons are done with different well-known configurations (capsule in shear flow and square-section channel), and by deepening the effect of the elastic constitutive law and capillary number on the membrane dynamics. Finally, to show the potential of this framework we introduce a new test case that describes the relaxation of a capsule in an opening channel. In order to increase the challenges of this test we study the influence of an internal nucleus, modeled as a hyperelastic solid, on the membrane evolution. Several numerical simulations of a 3D relaxation phenomenon are presented to provide characteristic shapes and curves related to the capsule deformations, while also modifying size and stiffness parameter of the nucleus.

physics.flu-dyn↗

arXiv2024

Extrapolated Shock Tracking: Bridging shock-fitting and embedded boundary methods

We propose a novel approach to approximate numerically shock waves. The method combines the unstructured shock-fitting approach developed in the last decade by some of the authors, with ideas coming from embedded boundary techniques. The numerical method obtained allows avoiding the re-meshing phase required by the unstructured fitting method, while guaranteeing accuracy properties very close to those of the fitting approach. This new method has many similarities with front tracking approaches, and paves the way to shock-tracking techniques truly independent on the data and mesh structure used by the flow solver. The approach is tested on several problems showing accuracy properties very close to those of more expensive fitting methods, with a considerable gain in flexibility and generality.

math.NA↗

arXiv2024

A high-order, fully well-balanced, unconditionally positivity-preserving finite volume framework for flood simulations

In this work, we present a high-order finite volume framework for the numerical simulation of shallow water flows. The method is designed to accurately capture complex dynamics inherent in shallow water systems, particularly suited for applications such as tsunami simulations. The arbitrarily high-order framework ensures precise representation of flow behaviors, crucial for simulating phenomena characterized by rapid changes and fine-scale features. Thanks to an {\it ad-hoc} reformulation in terms of production-destruction terms, the time integration ensures positivity preservation without any time-step restrictions, a vital attribute for physical consistency, especially in scenarios where negative water depth reconstructions could lead to unrealistic results. In order to introduce the preservation of general steady equilibria dictated by the underlying balance law, the high-order reconstruction and numerical flux are blended in a convex fashion with a well-balanced approximation, which is able to provide exact preservation of both static and moving equilibria. Through numerical experiments, we demonstrate the effectiveness and robustness of the proposed approach in capturing the intricate dynamics of shallow water flows, while preserving key physical properties essential for flood simulations.

math.NA↗

arXiv2024

Extrapolated DIscontinuity Tracking for complex 2D shock interactions

A new shock-tracking technique that avoids re-meshing the computational grid around the moving shock-front was recently proposed by the authors (Ciallella et al., 2020). The method combines the unstructured shock-fitting (Paciorri and Bonfiglioli,2009) approach, developed in the last decade by some of the authors, with ideas coming from embedded boundary methods. In particular, second-order extrapolations based on Taylor series expansions are employed to transfer the solution and retain high order of accuracy. This paper describes the basic idea behind the new method and further algorithmic improvements which make the extrapolated Discontinuity Tracking Technique (eDIT) capable of dealing with complex shock-topologies featuring shock-shock and shock-wall interactions occurring in steady problems. This method paves the way to a new class of shock-tracking techniques truly independent on the mesh structure and flow solver. Various test-cases are included to prove the potential of the method, demonstrate the key features of the methodology, and thoroughly evaluate several technical aspects related to the extrapolation from/onto the shock, and their impact on accuracy, and conservation.

math.NA↗

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

Semi-implicit Eulerian method for the fluid structure interaction of elastic membranes

In this paper we propose a novel and general approach to design semi-implicit methods for the simulation of fluid-structure interaction problems in a fully Eulerian framework. In order to properly present the new method, we focus on the two-dimensional version of the general model developed to describe full membrane elasticity. The approach consists in treating the elastic source term by writing an evolution equation on the structure stress tensor, even if it is nonlinear. Then, it is possible to show that its semi-implicit discretization allows us to add to the linear system of the Navier-Stokes equations some consistent dissipation terms that depend on the local deformation and stiffness of the membrane. Due to the linearly implicit discretization, the approach does not need iterative solvers and can be easily applied to any Eulerian framework for fluid-structure interaction. Its stability properties are studied by performing a Von Neumann analysis on a simplified one-dimensional model and proving that, thanks to the additional dissipation, the discretized coupled system is unconditionally stable. Several numerical experiments are shown for two-dimensional problems by comparing the new method to the original explicit scheme and studying the effect of structure stiffness and mesh refinement on the membrane dynamics. The newly designed scheme is able to relax the time step restrictions that affect the explicit method and reduce crucially the computational costs, especially when very stiff membranes are under consideration.

math.NA↗

arXiv2023

Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation

In the context of preserving stationary states, e.g. lake at rest and moving equilibria, a new formulation of the shallow water system, called Flux Globalization has been introduced by Cheng et al. (2019). This approach consists in including the integral of the source term in the global flux and reconstructing the new global flux rather than the conservative variables. The resulting scheme is able to preserve a large family of smooth and discontinuous steady state moving equilibria. In this work, we focus on an arbitrary high order WENO Finite Volume (FV) generalization of the global flux approach. The most delicate aspect of the algorithm is the appropriate definition of the source flux (integral of the source term) and the quadrature strategy used to match it with the WENO reconstruction of the hyperbolic flux. When this construction is correctly done, one can show that the resulting WENO FV scheme admits exact discrete steady states characterized by constant global fluxes. We also show that, by an appropriate quadrature strategy for the source, we can embed exactly some particular steady states, e.g. the lake at rest for the shallow water equations. It can be shown that an exact approximation of global fluxes leads to a scheme with better convergence properties and improved solutions. The novel method has been tested and validated on classical cases: subcritical, supercritical and transcritical flows.

math.NA↗

arXiv2022

An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.

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↗