arXiv subjects

Firas Dhaouadi

Publications and source records attributed to Firas Dhaouadi.

8 recordsLinked to original sources

arXiv2026

Energy consistent hyperbolic approximation for a class of fourth-order partial differential equations

In this article, we propose a novel hyperbolic relaxation system for a general class of fourth-order nonlinear partial differential equations arising in the modelling of thin film flows or the phase separation of binary mixtures. The approximations are constructed to dissipate energies which recover the energy (Lyapunov) functional of the limit equation when the relaxation parameters vanish. Using the relative energy framework, we prove the convergence of weak entropy solutions of the relaxation system to smooth solutions of the limit equation. We validate our analysis with a series of numerical examples for thin film equations and Cahn-Hilliard equations.

math.AP↗

arXiv2026

Analysis and structure-preserving discretization of the Heat-GLM system

In this paper, we study a prototype hyperbolic system arising from the coupling of Generalized-Lagrangian-Multiplier (GLM) curl-cleaning with linear acoustics. We first show that, in the absence of dissipation, the equations can be rigorously derived from an underlying variational principle and that the resulting system is symmetric-hyperbolic. Further analysis shows that it not only conserves the total energy but also admits a set of invariants consisting of quadratic combinations of differential operators applied to the state variables. The addition of a stiff relaxation source term allows the system to be extended to dissipative dynamics, describing, for example, Cattaneo-type heat transfer processes that are asymptotically compatible with the Fourier law in the stiff relaxation limit; in this case, the total energy and invariants are dissipated accordingly. A new semi-implicit compatible numerical scheme on staggered grids is developed to solve this system while exactly preserving its key properties at the discrete level. In particular, we prove that the scheme is asymptotic-preserving in the Fourier limit, with a convergence rate depending on the initial data. We also show that the scheme conserves exactly the total energy as well as all the invariants in the absence of relaxation, and dissipates them consistently in its presence. These findings are demonstrated on a set of representative test cases.

math.NA↗

arXiv2025

On hyperbolic approximations for a class of dispersive and diffusive-dispersive equations

We introduce novel approximate systems for dispersive and diffusive-dispersive equations with nonlinear fluxes. For purely dispersive equations, we construct a first-order, strictly hyperbolic approximation. Local well-posedness of smooth solutions is achieved by constructing a unique symmetrizer that applies to arbitrary smooth fluxes. Under stronger conditions on the fluxes, we provide a strictly convex entropy for the hyperbolic system that corresponds to the energy of the underlying dispersive equation. To approximate diffusive-dispersive equations, we rely on a viscoelastic damped system that is compatible with the found entropy for the hyperbolic approximation of the dispersive evolution. For the resulting hyperbolic-parabolic approximation, we provide a global well-posedness result. Using the relative entropy framework \cite{dafermos2005hyperbolic}, we prove that the solutions of the approximate systems converge to solutions of the original equations. The structure of the new approximate systems allows to apply standard numerical simulation methods from the field of hyperbolic balance laws. We confirm the convergence of our approximations even beyond the validity range of our theoretical findings on set of test cases covering different target equations. We show the applicability of the approach for strong nonlinear effects leading to oscillating or shock-layer-forming behavior.

math.AP↗

arXiv2025

Structure Preserving Finite Volume Schemes on Voronoi Grids: Curl Involution, Asymptotic Limit and Thermodynamics

We propose a new curl-free and thermodynamically compatible finite volume scheme on Voronoi grids to solve compressible heat conducting flows written in first-order hyperbolic form. The approach is based on the definition of compatible discrete curl-grad operators, exploiting the triangular nature of the dual mesh. We design a cell solver reminiscent of the nodal solvers used in Lagrangian schemes to discretize the evolution equation for the thermal impulse vector, and we demonstrate that the resulting numerical scheme ensures energy conservation, local non-negative entropy production, as well as asymptotic consistency with the classical Fourier law in the stiff relaxation limit. A novel technique is proposed to transfer residuals from the dual to the primal mesh as subfluxes, which eventually yields the construction of entropy compatible semi-discrete methods. The scheme and its properties are validated on a set of numerical test cases.

math.NA↗

arXiv2024

An Eulerian hyperbolic model for heat transfer derived via Hamilton's principle: analytical and numerical study

In this paper, we present a new model for heat transfer in compressible fluid flows. The model is derived from Hamilton's principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler--Lagrange equation. The governing system is shown to be hyperbolic. It is asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine--Hugoniot conditions and the Clausius--Duhem inequality reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases.

math.AP↗

arXiv2024

A first-order hyperbolic reformulation of the Cahn-Hilliard equation

In this paper we present a new first-order hyperbolic reformulation of the Cahn-Hilliard equation. The model is obtained from the combination of augmented Lagrangian techniques proposed earlier by the authors of this paper, with a classical Cattaneo-type relaxation that allows to reformulate diffusion equations as augmented first order hyperbolic systems with stiff relaxation source terms. The proposed system is proven to be hyperbolic and to admit a Lyapunov functional, in accordance with the original equations. A new numerical scheme is proposed to solve the original Cahn-Hilliard equations based on conservative semi-implicit finite differences, while the hyperbolic system was numerically solved by means of a classical second order MUSCL-Hancock-type finite volume scheme. The proposed approach is validated through a set of classical benchmarks such as spinodal decomposition, Ostwald ripening and exact stationary solutions.

math.NA↗

arXiv2024

An exactly curl-free finite-volume scheme for a hyperbolic compressible barotropic two-phase model

We present a new second order accurate structure-preserving finite volume scheme for the solution of the compressible barotropic two-phase model of Romenski et. al in multiple space dimensions. The governing equations fall into the wider class of symmetric hyperbolic and thermodynamically compatible (SHTC) systems and consist of a set of first-order hyperbolic partial differential equations (PDE). In the absence of algebraic source terms, the model is subject to a curl-free constraint for the relative velocity between the two phases. The main objective of this paper is, therefore, to preserve this structural property exactly also at the discrete level. The new numerical method is based on a staggered grid arrangement where the relative velocity field is stored in the cell vertexes while all the remaining variables are stored in the cell centers. This allows the definition of discretely compatible gradient and curl operators, which ensure that the discrete curl errors of the relative velocity field remain zero up to machine precision. A set of numerical results confirms this property also experimentally.

math.NA↗

arXiv2020

Stability theory for some scalar finite difference schemes : Validity of the modified equations approach

In this paper, we discuss some limitations of the modified equations approach as a tool for stability analysis for a class of explicit linear schemes to scalar partial derivative equations. We show that the infinite series obtained by Fourier transform of the modified equation is not always convergent and that in the case of divergence, it becomes unrelated to the scheme. Based on these results, we explain when the stability analysis of a given truncation of a modified equation may yield a reasonable estimation of a stability condition for the associated scheme. We illustrate our analysis by some examples of schemes namely for the heat equation and the transport equation.

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