Numerical Analysis of a Model of Two Phase Compressible Fluid Flow

Abstract

We consider a model of a binary mixture of two immiscible compressible fluids. We propose a numerical scheme and discuss its basic properties: stability, consistency, convergence. The convergence is established via the method of generalized weak solutions combined with the weak–strong uniqueness principle.

Appendix A

1.1 Useful Equality for the Diffusive Upwind Flux

Here we prove Lemma 5.5.

Proof

First, recalling the discrete operators defined in Sect. 5.1 we obtain by direct calculation that

Next, it is easy to get

Summing the two identities above we finish the proof of Lemma 5.5. $\square$

1.2 Existence of a Numerical Solution

Here we prove Lemma 5.9, that is the existence of a solution and positivity of density for the numerical method (5.6). We shall show the proof via a topological degree theory, which was reported in Gallouët et al. [13].

Theorem A.1

([13, Theorem A.1] Topological degree theory.) Let M and N be two positive integers. Let $C_1>\varepsilon >0$ and $C_2>0$ be real numbers. Let

$$\begin{aligned}&V = \{ (r,U) \in R ^M \times R ^N; \ r_i >0 \; \forall \; i=1,\dots , M \}, \\&W = \{ (r,U) \in R ^M \times R ^N; | U | \le C_2 \text { and } \varepsilon< r_i < C_1 \; \forall \; i=1, \dots , M \}. \end{aligned}$$

Let ${{\mathcal {F}}}$ be a continuous function mapping $V \times [0,1]$ to $R ^M \times R ^N$ and satisfying:

• 1. $f \in W$ if $f \in V$ satisfies $F(f,\zeta )={\mathbf {0}}$ for all $\zeta \in [0,1]$;

• 2. The equation ${{\mathcal {F}}}(f, 0)={\mathbf {0}}$ is a linear system with respect to f and admits a solution in W.

Then there exists an $f \in W$ such that ${{\mathcal {F}}}(f,1) ={\mathbf {0}}$.

Now we are ready to prove Lemma 5.9.

Proof of Lemma 5.9

The idea of the proof is to construct a mapping ${{\mathcal {F}}}$ that satisfies Theorem A.1. We begin with the definition of the spaces V and W

$$\begin{aligned}&V =\left\{ ( \varrho _h^k, U_h^k) \in Q_h\times {\mathfrak {Q}}_h,\ \varrho _h^k>0 \right\} ,\\&W =\left\{ ( \varrho _h^k, U_h^k ) \in Q_h\times {\mathfrak {Q}}_h,\ \Vert U_h^k\Vert \le C_2, \epsilon< \varrho _h^k <C_1 \right\} , \end{aligned}$$

where $U_h:=(\mathbf{u}_h, {\mathfrak {c}}_h) \in {\mathbf {V}}_h\times X_h=: {\mathfrak {Q}}_h$, $\varrho _h>c$ means $\varrho _K>c$ for all $K\in {\mathcal {T}}$, and the norm $\Vert U_h\Vert$ is given by $\Vert U_h\Vert \equiv \Vert \mathbf{u}_h\Vert _{L^6} + \Vert {\mathfrak {c}}_h\Vert _{L^6}$. Obviously, the dimensions of the spaces $Q_h$ and ${\mathfrak {Q}}_h$ are finite.

Next, for $\zeta \in [0,1]$ and $U^\star =(\mathbf{u}^\star , {\mathfrak {c}}^\star , \mu ^\star )$ we define the following mapping

$$\begin{aligned} {{\mathcal {F}}} : \ V \times [0,1]\rightarrow Q_h\times {\mathfrak {Q}}_h, \quad (\varrho _h^k,U_h^k, \zeta )\longmapsto ( \varrho ^\star , U^\star ) ={{\mathcal {F}}}(\varrho _h^k,U_h^k ,\zeta ), \end{aligned}$$

where $( \varrho ^\star , U^\star )$ is defined by:

(A.1a)

(A.1b)

(A.1c)

for any $\phi _h \in Q_h$ and $\varvec{\phi }_h \times \psi _h \in {\mathfrak {Q}}_h$, where $\varvec{\phi }_h=(\phi _{1,h}, \dots , \phi _{d,h})$, and the discrete Laplace operator is defined by the following equality

$$\begin{aligned} - \int _{\Omega } \Delta _h{\mathfrak {c}}_h\psi _h \,\mathrm{d} {x} = (1-\zeta ) \int _{\Omega } {\mathfrak {c}}_h\psi _h \,\mathrm{d} {x} + B({\mathfrak {c}}_h,\psi _h) . \end{aligned}$$

It is obvious that ${{\mathcal {F}}}$ is well defined and continuous since the values of $\varrho ^\star$ and $U^\star =(\mathbf{u}^\star , {\mathfrak {c}}^\star )$ can be determined by setting $\phi _h = 1_{K}$, $K\in {\mathcal {T}}$, in (A.1a), $\phi _{i,h}=1_\sigma , \sigma \in {\mathcal {E}}$ with $\phi _{j,h}=0$ for $j\ne i, \; i,j \in (1,\ldots , d)$ in (A.1b), and $\psi _h =1_{P}$, P being a degree of freedom of the space $X_h$ in (A.1c), respectively.

With the above definitions, we aim to show that both hypotheses of Theorem A.1 hold.

We first aim to prove that Hypothesis 1 of Theorem A.1 holds. To this end, we suppose $(\varrho _h^k,U_h^k ) \in Q_h\times {\mathfrak {Q}}_h$ is a solution to ${{\mathcal {F}}}(\varrho _h^k,U_h^k , \zeta )= {\mathbf {0}}$ for any $\zeta \in [0,1]$. Then the system (A.1) becomes

(A.2a)

(A.2b)

(A.2c)

Taking $\phi _h=1$ as a test function in (A.2a) we immediately obtain

$$\begin{aligned} \Vert \varrho _h^k\Vert _{L^1}= \int _{\Omega } \varrho _h^k \,\mathrm{d} {x} = \int _{\Omega } \varrho _h^{k-1} \,\mathrm{d} {x} \equiv M_0 >0. \end{aligned}$$

(A.3)

Further, following the proof of the energy stability (5.16) we know that

$$\begin{aligned}&D_t \int _{\Omega } \left( \frac{1}{2} \varrho _h\left|\widehat{\mathbf{u}_h^k} \right|^2 + \frac{1}{2} |\Vert {\mathfrak {c}}_h^k\Vert |_B^2+ \zeta F({\mathfrak {c}}_h^k) + \frac{1}{2} (1-\zeta ) |{\mathfrak {c}}_h^k|^2+ P(\varrho _h^k) \right) \,\mathrm{d} {x}+ \nu \Vert \nabla _h\mathbf{u}_h^k\Vert _{L^2}^2 \\&\quad + \zeta \eta \Vert \mathrm{div}_h\mathbf{u}_h^k\Vert _{L^2}^2 + \Vert D_t {\mathfrak {c}}_h^k + \zeta \mathbf{u}_h^k \cdot \nabla _h{\mathfrak {c}}_h^k \Vert _{L^2}^2 \le 0. \end{aligned}$$

Then we may apply the discrete Sobolev’s inequality stated in Lemma 5.4 to derive

$$\begin{aligned} \Vert U_h^k \Vert \equiv \Vert \mathbf{u}_h^k\Vert _{L^6} + \Vert {\mathfrak {c}}_h^k \Vert _{L^6} \le C_2, \end{aligned}$$

(A.4)

where $C_2>0$ depends on the initial data of the problem.

Next, let $K\in {\mathcal {T}}$ be such that $\varrho _K^k = \min _{L \in {\mathcal {T}}} \varrho _h^k|_L$. Now setting $\phi _h = 1_K$ and noticing we find

Thus $\varrho _h^k \ge \varrho _K^k \ge \frac{\varrho ^{k-1}_{K} }{1 + \Delta t\zeta \left|(\mathrm{div}_h\mathbf{u}_h^k)_K \right| } >0.$ Consequently, by virtue of (A.4) $\varrho _h^k > \epsilon$, where $\epsilon$ depends only on the data of the problem. Further, we get from (A.3) that $\varrho _h^k \le \frac{ M_0 }{\min _{K\in {\mathcal {T}}} |K|}$, which indicates the existence of $C_1>0$ such that $\varrho _h^k <C_1$. Therefore, Hypothesis 1 of Theorem A.1 is satisfied.

We also need to show that Hypothesis 2 of Theorem A.1 is satisfied. Let $\zeta =0$ then the system ${{\mathcal {F}}}(\varrho _h^k,U_h^k,0 )={\mathbf {0}}$ reads

(A.5a)

(A.5b)

(A.5c)

From (A.5a) it is obvious $\varrho _h^k = \varrho _h^{k-1}>0$. Substituting (A.5a) into (A.5b) we arrive at a linear system for $\mathbf{u}_h^k$ with a symmetric positive definite matrix. Thus (A.5b) admits a unique solution. Further, by noticing (A.5c) is a linear system of ${\mathfrak {c}}_h^k$ with a positive definite matrix, we know that it admits a unique solution ${\mathfrak {c}}_h^k$.

Consequently, Hypothesis 2 of Theorem A.1 is satisfied.

We have shown that both hypotheses of Theorem A.1 hold. Applying Theorem A.1 finishes the proof of Lemma 5.9. $\square$