On a New Spatial Discretization for a Regularized 3D Compressible Isothermal Navier–Stokes–Cahn–Hilliard System of Equations with Boundary Conditions

On properties of aggregated regularized systems of equations for a homogeneous multicomponent gas mixture

“…Moreover, the property z ∈ C([0, T]; L 2 (Ω)) and the formula∫ T 𝜕 t z(•, t), z ⟩ Ω) ∀T > 0follows from Gaewski et al24, Ch. IV, Theorem 1.17 and Remark 1 22. The function u 2 *  Q T (z, z) does not decrease for T≥0 and, summed up with 0.5||z(•, T)|| 2 L 2 (Ω) , is constant in T in equality(3.20), thus ||z(•, T)|| L 2 (Ω) does not increase for T≥0.…”

“…[13][14][15][16][17] Various regularized QGD and QHD systems of equations for binary mixtures in the absence of chemical reactions, including both non-homogeneous mixtures (when not only the densities but also the velocities and temperatures of the components are different) and homogeneous ones (i.e., with the common velocity and temperature of the mixture components), have been constructed, discretized and successfully tested in numerical simulations including those in previous works. 3,[18][19][20][21][22][23] This paper studies the properties of aggregated QGD and QHD systems of equations for the homogeneous multicomponent gas mixture which are essential as a mathematical basis for the success of the mentioned discretizations. This is carried out in the unified manner for the both systems.…”

“…Important mathematical results on the properties of their solutions were proved, in particular, in papers 13–17 . Various regularized QGD and QHD systems of equations for binary mixtures in the absence of chemical reactions, including both non‐homogeneous mixtures (when not only the densities but also the velocities and temperatures of the components are different) and homogeneous ones (i.e., with the common velocity and temperature of the mixture components), have been constructed, discretized and successfully tested in numerical simulations including those in previous works 3,18–23 …”

Correction to the Paper: An Energy Dissipative Spatial Discretization for the Regularized Compressible Navier-Stokes-Cahn-Hilliard System of Equations (In Math. Model. Anal., 25(1): 110–129, HTTPS://DOI.ORG/10.3846/MMA.2020.10577)

“…In this note, we correct the proof of Theorem 2 (p. 120-121) in [1] following our recent paper [2]. Below we exploit the notation from [1].…”

On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations

“…These methods are widely used for numerical solution of various applied problems. In the barotropic case, gas dynamics systems of equations with such regularizations were introduced and studied in [10][11][12][13], and their numerous applications to computer simulation were given for various 1D and 2D shallow water models [14][15][16][17][18][19], some 2D astrophysical problems [20] and the 2D and 3D compressible Navier-Stokes-Cahn-Hilliard models [21][22][23], etc. Despite of a lot of applications, almost nothing was known until recent years on rigorous theoretical conditions for stability of schemes with the QGD and QHD regularizations thus leading to additional time-consuming preliminary numerical experiments in order to choose the adequate parameters of schemes allowing to use larger values of ∆t.…”