Thermodynamically consistent spatial discretization of the one-dimensional regularized system of the Navier–Stokes–Cahn–Hilliard equations

An Energy Dissipative Spatial Discretization for the Regularized Compressible Navier-Stokes-Cahn-Hilliard System of Equations

“…In this discretization, the sought functions, namely, the density and velocity of the mixture and the concentration of one of the components are defined at nodes of one and the same grid. The corresponding 1D construction has recently been accomplished in [6]. The constructed 3D discretization is additionally well-balanced for equilibrium solutions and the potential body force.…”

“…In addition, we prove that the constructed discretization is wellbalanced for the potential body force. Also a version of this discretization more close to [6] is discussed. The last Section 4 is devoted to computer simulation of three 2D test problems.…”

“…In [6], to construct an energy dissipative 1D discretization, the following non-divergent form of terms ∇p + div Q in equation (2.2) and formula (2.4) has recently been applied…”

“…Hereafter the summation from 1 to 3 is assumed over the repeated indices i and j (and only over them). We discretize the components of regularized mass flux J k (x, t) on ω k * ,h , namely, the terms with ρ, u and Φ according to [27,Section 3.4] and the terms with C according to [6]:…”

“…Remark 3. Similarly to [6], the term s * l (s l ρ)(s l µ)δ l C can be replaced with µs * l (s l ρ)δ l C in equation (3.3) together with replacing the term δ (3.4). In this case, Theorem 1 remains valid.…”