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An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations
Journal of Computational Dynamics. 2020. Vol. 7. No. 2. P. 291–312.
Balashov V., Zlotnik A.
We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier-Stokes-Cahn-Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and keeping the component concentration inside the physically reasonable range (0,1)$. This is achieved, first, by discretizing the non-divergent potential form of terms responsible for the capillary effects and establishing the dissipativity of the discrete full energy. Second, a logarithmic (or the Flory-Huggins potential) form for the non-convex bulk free energy is used. The regularization of equations is accomplished to increase essentially the time step of the explicit discretization in time. We include 3D numerical results for two typical problems that confirm the theoretical predictions.
Teregulov T., Loubenets E. R., / Series Quantum Physics "arXiv". 2026. No. 2609.31472.
We develop a new three-party quantum secret-sharing (QSS) protocol based on a three-qubit W state. This protocol encodes the secret-sequence bits using X gates and employs randomly selected Hadamard operations and measurement bases to generate information and security-test rounds. We evaluate the efficiency of the proposed protocol and analyze its security against an internal adversary ...
Added: September 28, 2026
Levashev V., / Series arXiv "math". 2026. No. 2609.06010.
We prove that continuous A-bilinear pairings on the ring of Laurent series that are invariant under continuous automorphisms coincide, up to a constant, with the pairing given by the residue of a differential form over any commutative associative ring with identity. ...
Added: September 24, 2026
Pyatov P. N., Pivovarov P. A., / Series math "arxiv.org". 2026. No. 2609.06274.
We investigate a special ansats that allows for an iterative solution of the constant Yang-Baxter equation. Testing this ansatz, we construct four sequences of the constant R-matrices. In each sequence the R-matrices act on the tensor squares of vector spaces of linearly growing dimensions. Each R-matrix also depends on a single complex parameter.
By analyzing the ...
Added: September 24, 2026
Medvedev V., / Series arXiv "math". 2026.
We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit ...
Added: September 19, 2026
Poddiakov A., / Series Social Science Research Network "Social Science Research Network". 2026. No. 7437658.
Clarity of knowledge and reasoning is necessary in many cases. Yet vagueness in scientific thinking related to surprise, curiosity, "ability to engage with not-knowing" (de Freitas) and abductive reasoning is also a crucially important source of scientific creativity which supplements combinatorial logic when dealing with the already known. Starting from studies by C. S. Peirce ...
Added: September 15, 2026
Glutsyuk A., / Series arXiv "math". 2026.
B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, dθdτ=1ω(cosθ+B+Acosτ), which is known as ...
Added: September 8, 2026
Pochinka O., Shmukler V., / Series math.RT "arXiv:1808.06395 [math.RT]". 2026.
Anosov flows have a long and rich history, firstly motivated by the
study of geodesic flows in negative curvature surface by Anosov and Sinai.
Not every closed manifold admits an Anosov flow for well-known reasons:
the fundamental group of a 3-manifold M admitting an Anosov flow must
have exponential growth, and M must be universally covered by R3. Nevertheless,
there ...
Added: August 31, 2026
Shabanov N., Kuricheva O., Kurbatova J. et al., / Series Working Papers SSRN "Department of Economics Ca’ Foscari University of Venice". 2026.
The carbon balance of an ecosystem is the difference between Gross Primary Productivity (GPP) and Ecosystem Respiration (Reco) as expressed by Net Ecosystem Exchange (NEE). While remote sensing retrievals of GPP have reached maturity, Reco estimation remains underexplored and ultimately cast bias on NEE. Here we present an end-to-end multi-scale analysis of the mechanism of ...
Added: August 21, 2026
Ponomarenko A., / Series Computer Science "arxiv.org". 2025.
This paper addresses the challenge of merging hierarchical navigable small world (HNSW) graphs, a critical operation for distributed systems, incremental indexing, and database compaction. We propose three algorithms for this task: Naive Graph Merge (NGM), Intra Graph Traversal Merge (IGTM), and Cross Graph Traversal Merge (CGTM). These algorithms differ in their approach to vertex selection ...
Added: July 30, 2026
Loubenets E. R., / Series arxiv.org "quant-ph". 2026. No. 2607.18050.
In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers S1,S2≥1 of generalized quantum measurements at two sites. In the present article, we find analytically a new general locality condition sufficient for a nonseparable Werner state with a ...
Added: July 21, 2026
Bolbachan V., / Series math "arxiv.org". 2024.
Chow polylogarithms are some special functions arising in explicit description of the Beilinson regulator map. The most interesting functional equation for this function reflects its vanishing on the boundary in the Bloch's cycle complex. We show that this functional equation formally follows from more simple ones, namely skew-symmetry, functoriality and multiplicativity.
To prove this, we study ...
Added: July 16, 2026
Bolbachan V., / Series math "arxiv.org". 2024.
Let K be a field of characteristic zero. We prove that its motivic cohomology in degree m−1 and weight m is rationally isomorphic to the cohomology of the polylogarithmic complex. This gives a partial extension of A. Suslin theorem describing the indecomposable K3 of a field. ...
Added: July 16, 2026
Panov V., Ryabchenko A., / Series arXiv "stat.ME". 2026. No. 2607.05048.
This paper investigates the problem of statistical inference for a mixture distribution consisting of a discrete and a continuous component, with a particular focus on the class of rational-infinitely divisible distributions. We consider non-parametric estimation of both components of the mixture as well as the quasi-L{é}vy measure, assuming that the mixture belongs to the class ...
Added: July 9, 2026
Piontkovski D., / Series arXiv "math". 2026.
A noncommutative projective variety is defined, following Artin and Zhang, by a graded coherent algebra 𝐴. The category of coherent sheaves is then the quotient qgr(𝐴) of the category of finitely presented graded modules by the subcategory of torsion modules. We consider the categorical and polynomial entropies of the Serre twist, that is, of the ...
Added: June 23, 2026
Piontkovski D., / Series arXiv "math". 2025.
If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then its Jacobian matrix is invertible. This refines a theorem of Gerstenhaber and partially answers a question posed ...
Added: June 23, 2026
A. A. Zlotnik, T. A. Lomonosov, Computational Mathematics and Mathematical Physics (Germany) 2022 Vol. 62 No. 11 P. 1817–1837
We study an explicit two-level finite difference scheme on staggered meshes, with a quasi-hydrodynamic regularization, for 1D barotropic gas dynamics equations. We derive both necessary conditions and sufficient conditions close to them, for $L^2$-dissipativity of solutions to the Cauchy problem linearized on a constant solution, for any background Mach number $M$. We apply the spectral approach and analyze matrix ...
Added: October 18, 2022
Zlotnik A., Lomonosov T., Математическое моделирование 2021 Т. 33 № 5 С. 16–34
We study explicit two-level finite-difference schemes on staggered meshes for two known regularizations of 1D barotropic gas dynamics equations including schemes with discretizations in x that possess the dissipativity property with respect to the total energy. We derive criterions of L^2-dissipativity in the Cauchy problem for their linearizations at a constant solution with zero background ...
Added: April 1, 2021
Balashov V., Zlotnik A., Journal of Scientific Computing 2021 Vol. 86 Article 33
We construct a new spatial finite-difference discretization for a regularized 3D Navier-Stokes-Cahn-Hilliard system of equations. The system {can be attributed to phase field type models} and describes flows of a viscous
compressible isothermal two-component two-phase fluid with surface effects; the potential body force is also taken into account. In the discretization, the main sought functions are defined on one ...
Added: December 20, 2020
Balashov V., Zlotnik A., Mathematical Modelling and Analysis 2020 Vol. 25 No. 1 P. 110–129
We consider the regularized 3D Navier-Stokes-Cahn-Hilliard equations describing isothermal flows of viscous compressible two-component fluids with interphase effects. We construct for them a new energy dissipative finite-difference discretization in space, i.e., with the non-increasing total energy in time. This property is preserved in the absence of a regularization. In addition, the discretization is well-balanced for equilibrium flows and the ...
Added: October 27, 2019
Bellettini G., Bertini L., Mariani M. et al., SIAM Journal on Mathematical Analysis 2012 Vol. 44 No. 5 P. 3458–3480
We consider the Cahn-Hilliard equation in one space dimension with scaling parameter epsilon, i.e., u(t) = (W'(u) - epsilon(2)u(xx))(xx), where W is a nonconvex potential. In the limit epsilon down arrow 0, under the assumption that the initial data are energetically well prepared, we show the convergence to a Stefan problem. The proof is based ...
Added: October 8, 2018
Balashov V., Savenkov E., Zlotnik A., Russian Journal on Numerical Analysis and Mathematical Modelling 2019 Vol. 34 No. 1 P. 1–13
A numerical algorithm is proposed for modeling two-component viscous compressible isothermal flows with surface effects in 3D domains of complex shape with a voxel representation of geometry. The regularized system of Navier-Stokes-Kahn-Hillard equations is used as a basic mathematical model. Modeling of the drop spreading on a flat substrate and the displacement of one liquid by ...
Added: September 28, 2018
Balashov V., Zlotnik Alexander, Savenkov E., Russian Journal on Numerical Analysis and Mathematical Modelling 2017 Vol. 32 No. 6 P. 347–358
A study of a regularized quasi-hydrodynamic system of equations for the two-component isothermal mixture with the diffuse interface is accomplished. Under general assumptions on the Helmholtz free energy of the mixture, the energy balance equation with non-positive energy production and its corollary, the law of non-increasing total energy, are derived. The necessary and sufficient conditions ...
Added: February 26, 2017
Zlotnik A., Gavrilin V., Mathematical Modelling and Analysis 2011 Vol. 16 No. 4 P. 509–526
A quasi-gasdynamic system of equations with a mass force and a heat source is well known in the case of the perfect polytropic gas. In the paper, the system is generalized to the case of general equations of gas state satisfying thermodynamic stability conditions. The entropy balance equation is studied. The validity of the non-negativity ...
Added: July 5, 2012