arXiv subjects

Mária Lukáčová-Medvid'ová

Publications and source records attributed to Mária Lukáčová-Medvid'ová.

At least 19 recordsLinked to original sources

arXiv2026

Fully Discrete Active Flux Methods for the Euler Equations: Exploring Different Reconstructions and New Multi-Dimensional Evolution Operators

We present new fully discrete Cartesian grid Active Flux methods for the two-dimensional Euler equations of gas dynamics. A key component of these methods is the evolution operator used to evolve point-value degrees of freedom. Therefore, we derive several new, third-order accurate, multidimensional evolution operators for point-value updates of the linearized Euler equations using the method of bicharacteristics. We also investigate how the reconstruction and choice of approximate evolution operator affect the accuracy and stability of the resulting methods.

math.NA↗

arXiv2026

Error analysis of an asymptotic-preserving, energy-stable finite volume method for barotropic Euler equations

We propose and analyse an energy-stable and asymptotic-preserving finite volume scheme for the compressible Euler system. Using the relative energy framework, we establish rigorous error estimates that yield convergence of the numerical solutions in two distinct regimes. For a fixed Mach number $\varepsilon>0$, we derive error estimates between the numerical solutions and a strong solution of the compressible Euler system that are uniform with respect to the discretisation parameters, ensuring convergence as the underlying mesh is refined. In the low Mach number regime, we analyse the error between the numerical solutions and a strong solution of the incompressible Euler system and obtain asymptotic error estimates that are uniform in $\varepsilon$ and the discretisation parameters. These results imply convergence of the numerical solutions toward a strong solution of the incompressible Euler system as $\varepsilon$, and the discretisation parameters simultaneously tend to zero. Numerical experiments are presented to validate the theoretical analysis.

math.NA↗

arXiv2026

Selection criteria for the compressible Euler equations: numerical study

We numerically investigate selection criteria for dissipative weak solutions of the compressible Euler equations. Using the Kelvin-Helmholtz problem as a benchmark, we compare four standard numerical schemes - the viscous finite volume (VFV) method, the discrete velocity Boltzmann finite volume (DVBFV) method, the Runge-Kutta discontinuous Galerkin (RKDG) method, and the discontinuous Galerkin spectral element method (DGSEM), and demonstrate that each may converge to a different dissipative weak solution. We evaluate these solutions with respect to several selection criteria based on entropy production, total energy, and energy defect, and examine the sensitivity of selection functionals to the numerical diffusion inherent in each scheme.

math-ph↗

arXiv2026

Oscillatory approximations and maximum entropy principle for the Euler system of gas dynamics

We show that the measure-valued solutions of the Euler system of gas dynamics generated by oscillatory sequences of consistent approximations violate the principle of maximal entropy production formulated by Dafermos. Numerical results illustrate that solutions obtained by standard numerical methods may be oscillatory and thus do not comply with the Dafermos criterion.

math.AP↗

arXiv2026

Weak-strong uniqueness for bi-fluid compressible system with algebraic closure

We consider a real two-fluid system of compressible viscous fluids with a common velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The existence of global-in-time finite energy weak solutions to this system is known since the work of Novotný and Pokorný [Arch. Rational Mech. Anal., 2020]. On the other hand, existence of local-in-time strong solutions is due to Piasecki and Zatorska [J. Math Fluid Mech., 2022]. In this paper, we establish the weak-strong uniqueness principle using the relative entropy method. In sharp contrast to the two-phase model of Baer-Nunziato type, the volume fraction of phase $+$ obeys a transport equation with an additional nonlinear term. This gives rise to troublesome terms in the relative entropy inequality. We are able to close the estimate by making an elaborate use of the structure of the system.

math.AP↗

arXiv2025

Convergence of a Hyperbolic Thermodynamically Compatible Finite Volume scheme for the Euler equations

We study the convergence of a novel family of thermodynamically compatible schemes for hyperbolic systems (HTC schemes) in the framework of dissipative weak solutions, applied to the Euler equations of compressible gas dynamics. Two key novelties of our method are i) entropy is treated as one of the main field quantities and ii) the total energy conservation is a consequence of compatible discretization and application of the Abgrall flux.

math.NA↗

arXiv2025

Provably fully discrete energy-stable and asymptotic-preserving scheme for barotropic Euler equations

We develop structure-preserving finite volume schemes for the barotropic Euler equations in the low Mach number regime. Our primary focus lies in ensuring both the asymptotic-preserving (AP) property and the discrete entropy stability. We construct an implicit-explicit (IMEX) method with suitable acoustic/advection splitting including implicit numerical diffusion that is independent of the Mach number. We prove the positivity of density, the entropy stability, and the asymptotic consistency of the fully discrete numerical method rigorously. Numerical experiments for benchmark problems validate the structure-preserving properties of the proposed method.

math.NA↗

arXiv2025

A note on relations between convexity and concavity of thermodynamic functions

The paper is concerned with proving the equivalence of convexity or concavity properties of thermodynamic functions, such as energy and entropy, depending on different sets of variables. These variables are the basic thermodynamic state variables, specific state variables or the densities of state variables that are used in continuum mechanics. We prove results for transformations of variables and functions in conjunction with convexity properties. We are concerned with convexity, strict convexity, positive definite Hessian matrices and the analogous forms of concavity. The main results are equivalence relations for these properties between functions. These equivalences are independent of the equations of state since they only use general properties of them. The results can be used for instance to easily prove that the entropy density function for the Euler equations in conservative variables in three space dimensions is strictly concave or even has a negative definite Hessian matrix. Further, we show how various equations of state imply these properties and how these properties are relevant to mathematical analysis.

math.AP↗

arXiv2025

Numerical Study of Random Kelvin-Helmholtz Instability

In this paper, we study random dissipative weak solutions of the compressible Euler equations in the Kelvin-Helmholtz (KH) instability. Motivated by the fact that weak entropy solutions are not unique and can be viewed as inviscid limits of Navier-Stokes flows, we take a statistical approach following ideas from turbulence theory. Our aim is to identify solution features that remain consistent across different realizations and mesh resolutions. For this purpose, we compute stable numerical solutions using a stochastic collocation method implemented with the help of a fifth-order alternative weighted essentially non-oscillatory (A-WENO) scheme and seventh-order central weighted essentially non-oscillatory (CWENO) interpolation in the random space. The obtained solutions are averaged over several embedded uniform grids, resulting in Cesáro averages, which are studied using stochastic tools. The analysis includes Reynolds stress and energy defects, probability density functions of averaged quantities, and reduced-order representations using proper orthogonal decomposition. The presented numerical experiments illustrate that random KH instabilities can be systematically described using statistical methods, averaging, and reduced-order modeling, providing a robust methodology for capturing the complex and chaotic dynamics of inviscid compressible flows.

math.NA↗

arXiv2025

An asymptotic preserving scheme satisfying entropy stability for the barotropic Euler system

In this paper we study structure-preserving numerical methods for low Mach number barotropic Euler equations. Besides their asymptotic preserving properties that are crucial in order to obtain uniformly consistent and stable approximations of the Euler equations in their singular limit as the Mach number approaches zero, our aim is also to preserve discrete entropy stability. Suitable acoustic/advection splitting approach combined with time implicit-explicit approximations are used to achieve the asymptotic preserving property. The entropy stability of different space discretisation strategies is studied for different values of Mach number and is validated by the numerical experiments.

math.NA↗

arXiv2025

Maximal dissipation and well-posedness of the Euler system of gas dynamics

We show that any dissipative (measure-valued) solution of the compressible Euler system that complies with Dafermos' criterion of maximal dissipation is necessarily an admissible weak solution. In addition, we propose a simple, at most two step, selection procedure to identify a unique semigroup solution in the class of dissipative solutions to the Euler system. Finally, we introduce a refined version of Dafermos' criterion yielding a unique solution of the problem for any finite energy initial data.

math.AP↗

arXiv2024

Asymptotic preserving finite volume method for the compressible Euler equations: analysis via dissipative measure-valued solutions

We propose and analyze a new asymptotic preserving (AP) finite volume scheme for the multidimensional compressible barotropic Euler equations to simulate low Mach number flows. The proposed scheme uses a stabilized upwind numerical flux, with the stabilization term being proportional to the stiff pressure gradient, and we prove its conditional energy stability and consistency. Utilizing the concept of dissipative measure-valued (DMV) solutions, we rigorously illustrate the AP properties of the scheme for well-prepared initial data. In particular, we prove that the numerical solutions will converge weakly to a DMV solution of the compressible Euler equations as the mesh parameter vanishes, while the Mach number is fixed. The DMV solutions then converge to a classical solution of the incompressible Euler system as the Mach number goes to zero. Conversely, we show that if the mesh parameter is kept fixed, we obtain an energy stable and consistent finite-volume scheme approximating the incompressible Euler equations as the Mach number goes to zero. The numerical solutions generated by this scheme then converge weakly to a DMV solution of the incompressible Euler system as the mesh parameter vanishes. Invoking the weak-strong uniqueness principle, we conclude that the DMV solution and classical solution of the incompressible Euler system coincide, proving the AP property of the scheme. We also present an extensive numerical case study in order to illustrate the theoretical convergences, wherein we utilize the techniques of K-convergence.

math.NA↗

arXiv2024

Monte Carlo method and the random isentropic Euler system

We show several results on convergence of the Monte Carlo method applied to consistent approximations of the isentropic Euler system of gas dynamics with uncertain initial data. Our method is based on combination of several new concepts. We work with the dissipative weak solutions that can be seen as a universal closure of consistent approximations. Further, we apply the set-valued version of the Strong law of large numbers for general multivalued mapping with closed range and the Komlós theorem on strong converge of empirical averages of integrable functions. Theoretical results are illustrated by a series of numerical simulations obtained by an unconditionally convergent viscosity finite volume method combined with the Monte Carlo method.

math.NA↗

arXiv2024

Convergence and error estimates of a penalization finite volume method for the compressible Navier-Stokes system

In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we present an elegant alternative to a direct, but rather technical, analysis of variational crimes by means of the penalty approach. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.

math.NA↗

arXiv2023

An implicit-explicit solver for a two-fluid single-temperature model

We present an implicit-explicit finite volume scheme for two-fluid single-temperature flow in all Mach number regimes which is based on a symmetric hyperbolic thermodynamically compatible description of the fluid flow. The scheme is stable for large time steps controlled by the interface transport and is computational efficient due to a linear implicit character. The latter is achieved by linearizing along constant reference states given by the asymptotic analysis of the single-temperature model. Thus, the use of a stiffly accurate IMEX Runge Kutta time integration and the centered treatment of pressure based quantities provably guarantee the asymptotic preserving property of the scheme for weakly compressible Euler equations with variable volume fraction. The properties of the first and second order scheme are validated by several numerical test cases.

math.NA↗

arXiv2022

Existence of dissipative solutions to the compressible Navier-Stokes system with potential temperature transport

We introduce dissipative solutions to the compressible Navier-Stokes system with potential temperature transport motivated by the concept of Young measures. We prove their global-in-time existence by means of convergence analysis of a mixed finite element-finite volume method. If a classical solution to the compressible Navier-Stokes system with potential temperature transport exists, we prove the strong convergence of numerical solutions. Our results hold for the full range of adiabatic indices including the physically relevant cases in which the existence of global-in-time weak solutions is open.

math.NA↗

arXiv2021

Convergence of a stochastic collocation finite volume method for the compressible Navier-Stokes system

We propose a stochastic collocation method based on the piecewise constant interpolation on the probability space combined with a finite volume method to solve the compressible Navier-Stokes system at the nodal points. We show convergence of numerical solutions to a statistical solution of the Navier-Stokes system on condition that the numerical solutions are bounded in probability. The analysis uses the stochastic compactness method based on the Skorokhod/Jakubowski representation theorem and the criterion of convergence in probability due to Gyöngy and Krylov.

math.NA↗

arXiv2021

A multi-scale method for complex flows of non-Newtonian fluids

We introduce a new heterogeneous multi-scale method for the simulation of flows of non-Newtonian fluids in general geometries and present its application to paradigmatic two-dimensional flows of polymeric fluids. Our method combines micro-scale data from non-equilibrium molecular dynamics (NEMD) with macro-scale continuum equations to achieve a data-driven prediction of complex flows. At the continuum level, the method is model-free, since the Cauchy stress tensor is determined locally in space and time from NEMD data. The modelling effort is thus limited to the identification of suitable interaction potentials at the micro-scale. Compared to previous proposals, our approach takes into account the fact that the material response can depend strongly on the local flow type and we show that this is a necessary feature to correctly capture the macroscopic dynamics. In particular, we highlight the importance of extensional rheology in simulating generic flows of polymeric fluids.

physics.flu-dyn↗