arXiv subjects

Fabio Leotta

Publications and source records attributed to Fabio Leotta.

5 recordsLinked to original sources

arXiv2026

Justification and structure- and asymptotic-preserving discretizations of a hyperbolized Cahn-Hilliard equation

We study a hyperbolic approximation ("hyperbolization") of the Cahn-Hilliard (CH) equation, originally proposed by Dhaouadi, Dumbser, and Gavrilyuk (2025, DOI: 10.1098/rspa.2024.0606) and study its convergence towards the CH model in a relaxation limit both via formal asymptotic expansions and, for a slightly modified approximation, via the relative energy framework. Moreover, we develop energy-stable semidiscretizations of the CH equation and of this hyperbolization using upwind summation-by-parts operators in space. Subsequently, we combine them with (additive) implicit-explicit (IMEX) Runge-Kutta methods based on a convex-concave splitting. We show that the resulting method is asymptotic preserving, i.e., it converges in the limit of the relaxation parameter to a stable discretization of the original CH equation. The choice of the necessary parameters is guided by the a priori error estimate based on the relative energy framework.

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arXiv2026

Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System

The Navier-Stokes-Cahn-Hilliard (NSCH) system governs the diffuse-interface dynamics of two incompressible and immiscible fluids. We consider a relaxation approximation of the NSCH system that is composed by a system of first-order hyperbolic balance laws and second-order elliptic operators. We prove first that the solutions of an initial boundary value problem for the approximation recover the limiting NSCH system for vanishing relaxation parameters. To cope with the singular limit we exploit the fact that the approximate solutions dissipate an almost quadratic energy, and employ the relative entropy-framework. In the second part of the work we provide numerical evidence for the analytical results, even in flow regimes not covered by the assumptions needed for the theoretical results. Using a novel marker-and-cell conservative finite-difference approach for both the approximation and the limit system, we are able to compute physically relevant interfacial flow problems including Ostwald ripening and high-velocity flow.

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arXiv2025

Conditional a priori error estimates of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D

We derive conditional a priori error estimates of a wide class of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D via the verification of weak consistency and entropy stability, as recently proposed by Bressan et al.~\cite{BressanChiriShen21}. Convergence in $L^\infty L^1$ with rate $h^{1/3}$ is obtained under a time step restriction $\tau\leq ch$, provided the following conditions hold: the exact solution is piecewise Lipschitz continuous, its (finitely many and isolated) shock curves can be traced with precision $h^{2/3}$ and, outside of these shock tracing tubular neighborhoods the numerical solution -- assumed to be uniformly small in BV -- has oscillation strength $h$ across each mesh cell and cell boundary.

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arXiv2024

A priori error estimates of Runge-Kutta discontinuous Galerkin schemes to smooth solutions of fractional conservation laws

We give a priori error estimates of second order in time fully explicit Runge-Kutta discontinuous Galerkin schemes using upwind fluxes to smooth solutions of scalar fractional conservation laws in one space dimension. Under the time step restrictions $τ\leq c h$ for piecewise linear and $τ\lesssim h^{4/3}$ for higher order finite elements, we prove a convergence rate for the energy norm $\|\cdot\|_{L^\infty_tL^2_x}+|\cdot|_{L^2_tH^{λ/2}_x}$ that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.

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arXiv2021

Second-order accurate normal reconstruction from volume fractions on unstructured meshes with arbitrary polyhedral cells

This paper introduces a novel method for the efficient second-order accurate computation of normal fields from volume fractions on unstructured polyhedral meshes. Locally, i.e. in each mesh cell, an averaged normal is reconstructed by fitting a plane in a least squares sense to the volume fraction data of neighboring cells while implicitly accounting for volume conservation in the cell at hand. The resulting minimization problem is solved approximately by employing a Newton-type method. Moreover, applying the Reynolds transport theorem allows to assess the regularity of the derivatives. Since the divergence theorem implies that the volume fraction can be cast as a sum of face-based quantities, our method considerably simplifies the numerical procedure for applications in three spatial dimensions while demonstrating an inherent ability to robustly deal with unstructured meshes. We discuss the theoretical foundations, regularity and appropriate error measures, along with the details of the numerical algorithm. Finally, numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes are presented, where we obtain second-order convergence for the normal alignment and symmetric volume difference. Moreover, the findings are substantiated by completely new deep insights into the minimization procedure.

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