# THE SECOND-ORDER STABILIZED GAUGE-UZAWA METHOD FOR INCOMPRESSIBLE FLOWS WITH VARIABLE DENSITY

• Kim, Taek-cheol (Department of Mathematics, Kangwon National University) ;
• Pyo, Jae-Hong (Department of Mathematics, Kangwon National University)
• Accepted : 2019.03.27
• Published : 2019.03.30

#### Abstract

The Navier-Stokes equations with variable density are challenging problems in numerical analysis community. We recently built the 2nd order stabilized Gauge-Uzawa method [SGUM] to solve the Navier-Stokes equations with constant density and have estimated theoretically optimal accuracy. Also we proved that SGUM is unconditionally stable. In this paper, we apply SGUM to the Navier-Stokes equations with nonconstant variable density and find out the stability condition of the algorithms. Because the condition is rather strong to apply to real problems, we consider Allen-Cahn scheme to construct unconditionally stable scheme.

#### File FIGURE 1. A low Atwood ratio problem of Algorithm 1 with finite element (P1, P1, P1) for (u, p, ρ) at Re = 1000(density contours 1.4 ≤ ρ ≤ 1.6) FIGURE 2. A low Atwood ratio problem of Algorithm 1 with finite element (P1, P1, P1) for (u, p, ρ) at Re =5000(density contours 1.4 ≤ ρ ≤ 1.6) FIGURE 3. A high Atwood ratio problem of Algorithm 1 with finite element (P1, P1, P1) for (u, p, ρ) at Re =1000(density contours 2 ≤ ρ ≤ 4) FIGURE 4. A low Atwood ratio problem of Algorithm 4 with finite element (P1, P1, P1) for (u, p, ρ) at Re =1000(density contours 1.4 ≤ ρ ≤ 1.6) FIGURE 5. A low Atwood ratio problem of Algorithm 4 with finite element (P1, P1, P1) for (u, p, ρ) at Re =5000(density contours 1.4 ≤ ρ ≤ 1.6)

TABLE 2. Error and convergence rate of Algorithm 2 with finite element (P2, P1, P1) for (u, p, ρ), μ = 1 and τ = 0:1 × h TABLE 1. Error and convergence rate of Algorithm 1 with finite element (P2, P1, P1) for (u, p, ρ), μ = 1 and τ = 0.1 × h TABLE 3. Error and convergence rate of Alrorithm 3 with finite element (P2, P1, P1) for (u, p, ρ), μ = 1 and τ = 0:1 × h #### Acknowledgement

Supported by : Kangwon National University

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