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# Numerical study of phase transition in van der Waals fluid

• * Corresponding author: Chang Liu
The first author is supported by NSFC grant No.11201322, the third author is supported by NSFC grant No.11671027.
• In this article, we use a relaxation scheme for conservation laws to study liquid-vapor phase transition modeled by the van der Waals equation, which introduces a small parameter $ε$ and a new variable. We solve the relaxation system in Lagrangian coordinates for one dimension and solve the system in Eulerian coordinates for two dimension. A second order TVD Runge-Kutta splitting scheme is used in time discretization and upwind or MUSCL scheme is used in space discretization. The long time behavior of the fluid is numerically investigated. If the initial data belongs to elliptic region, the solution converges to two Maxwell states. When the initial data lies in metastable region, the solution either remains in the same phase or converges to the Maxwell states depending to the initial perturbation. If the initial state is in the stable region, the solution remains in that region for all time.

Mathematics Subject Classification: Primary: 76T10, 35L65, 65M06; Secondary: 65D30.

 Citation: • • Figure 1.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.1 e^{-4x^{2}})^{T}$, $\epsilon = 0.0025$, 1025 grid points. $V(x)$ at various time T. Elliptic region is between the solid lines, and dashed lines represent the Maxwell states.

Figure 2.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-8}$, 1025 grid points.

Figure 3.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 0.005$, 1025 grid points.

Figure 4.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 0.025$, 1025 grid points.

Figure 5.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.9}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 0.05$, 1025 grid points.

Figure 6.  $(V_{0}, u_{0})^{T} = (\frac{1}{1.6}, 0.1 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-8}$, 1025 grid points.

Figure 7.  $(V_{0}, u_{0})^{T} = (\frac{1}{1.6}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-4}$, 1025 grid points.

Figure 8.  $(V_{0}, u_{0})^{T} = (\frac{1}{0.6+0.4\sin(x)}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-8}$, 1025 grid points.

Figure 9.  $(V_{0}, u_{0})^{T} = (\frac{1}{1.7}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-8}$, 1025 grid points.

Figure 10.  $(V_{0}, u_{0})^{T} = (\frac{1}{1.7+0.1\cos(x)}, 0.5 e^{-4x^{2}})^{T}$, $\epsilon = 10^{-8}$, 1025 grid points.

Figure 11.  $(\rho_{0}, u_{0}, v_{0})^{T} = (1.0, 0.01 e^{-4(x^{2}+y^{2})}, 0.01 e^{-4(x^{2}+y^{2})})^{T}$, $\epsilon = 0.002$, $256 \times 256$ grid points.

Figure 12.  $(\rho_{0}, u_{0}, v_{0})^{T} = (1.6, 0.01 e^{-4(x^{2}+y^{2})}, 0.01 e^{-4(x^{2}+y^{2})})^{T}$, $\epsilon = 0.002$, $256 \times 256$ grid points.

Figure 13.  $(\rho_{0}, u_{0}, v_{0})^{T} = (1.6, 0.5 e^{-4(x^{2}+y^{2})}, 0.5 e^{-4(x^{2}+y^{2})})^{T}$, $\epsilon = 0.002$, $256 \times 256$ grid points.

Figure 14.  $(\rho_{0}, u_{0}, v_{0})^{T} = (1.7, 0.01 e^{-4(x^{2}+y^{2})}, 0.01 e^{-4(x^{2}+y^{2})})^{T}$, $\epsilon = 0.002$, $256 \times 256$ grid points.

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