On numerical approximations to fluid–structure interactions involving compressible fluids

Abstract

In this paper we introduce a numerical scheme for fluid–structure interaction problems in two or three space dimensions. A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions.

A: Appendix

1.1 A.1 Proof of Theorem 3: existence of a numerical solution

We aim to prove Theorem 3 for the existence of a numerical solution. Before that let us first introduce an abstract theorem, see [30, Theorem A.1].

Theorem 6

([30, Theorem A.1]) Let M and N be positive integers. Let $C_2>\epsilon >0$ and $C_1>0$ be real numbers. Let V and W be defined as follows:

where the notation $x > c$ means that each component of x is greater than c, and $\Vert \cdot \Vert$ is a norm defined over $R ^N$. Let F be a continuous function from $V \times [0,1]$ to $R ^M \times R ^N$ satisfying:

• 1. $\forall \, \zeta \in [0,1]$, if $v \in V$ is such that $F(v,\zeta )=0$ then $v \in W$;

• 2. The equation $F (v, 0)=0$ is a linear system on v and has a solution in W.

Then there exists at least a solution $v \in W$ such that $F(v,1) = 0$.

Now we are ready to show Theorem 3.

Proof

Let us denote , and define

It is obvious that the degrees of freedom of the spaces $Q_{h,\tau }$ and $\mathbb {Q}$ are finite. Indeed, the space $Q_{h,\tau }$ can be identified by the set of values $\varrho _K$ for all $K \in \mathcal {T}_{h,\tau }^{k}$, therefore $Q_{h,\tau }\subset \mathbb {R}^M$, where M is the total number of elements of $\mathcal {T}_{h,\tau }^{k}$. Analogously, $\mathbb {Q}\subset \mathbb {R}^N$, where N is the sum of d times degrees of freedom of $\mathcal {E}^k$ and the degrees of freedom of $\Sigma$. Let us consider the mapping

$$\begin{aligned} F : \ V \times [0,1]\longrightarrow Q_h\times \mathbb {Q}. \qquad (\varrho _{h,\tau }^k,U_{h,\tau }^k , \zeta )\longmapsto ( \varrho ^\star , U^\star ) =F(\varrho _{h,\tau }^k,U_{h,\tau }^k ,\zeta ), \end{aligned}$$

where $( \varrho ^\star , U^\star ) \in Q_{h,\tau }\times \mathbb {Q}$ is such that

$$\begin{aligned}&\int _{\Omega _{h,\tau }} \varrho ^\star {\varphi }_{h,\tau } \,\mathrm{d} {x} = \int _{\Omega _{h,\tau }} \frac{\varrho _{h,\tau }^k -\varrho _{h,\tau }^{k-1}\circ {X}_k^{k-1}\mathcal {F}_k^{k-1}}{\tau } {\varphi }_{h,\tau } \,\mathrm{d} {x}\nonumber \\&\qquad + \zeta \int _{\Omega _{h,\tau }} \mathrm{div}^{\mathrm{up}}_{h,\tau }( \varrho _{h,\tau }^{k}, \mathbf {v}_{h,\tau }^{k} ){\varphi }_{h,\tau } \,\mathrm{d} {x} ; \end{aligned}$$

(A.1a)

$$\begin{aligned}&\int _{\Omega _{h,\tau }} U^\star \cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} \nonumber \\&\quad = \int _{\Omega _{h,\tau }} \frac{\varrho _{h,\tau }^k \Pi _\mathcal {T}[\mathbf {u}_{h,\tau }^{k}] - (\varrho _{h,\tau }^{k-1}\Pi _\mathcal {T}[\mathbf {u}_{h,\tau }^{k-1}]) \circ {X}_k^{k-1}\mathcal {F}_k^{k-1}}{\tau } \cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x}\nonumber \\&\qquad + \int _{\Sigma } \frac{ z_{h,\tau }^{k} - z_{h,\tau }^{k-1} }{\tau }{\psi }_{h,\tau } \mathrm{d}r + \int _{\Sigma } \Delta \eta _{h,\tau }^{k} \Delta {\psi }_{h,\tau } \mathrm{d}r \nonumber \\&\qquad - \int _{\Omega _{h,\tau }} \varrho _{h,\tau }^k \mathbf {f}_\tau ^{k}\cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} + \int _{\Sigma } g_\tau ^{k} {\psi }_{h,\tau } \mathrm{d}r\nonumber \\&\qquad +\zeta \int _{\Omega _{h,\tau }} \mathrm{div}^{\mathrm{up}}_{h,\tau }(\varrho _{h,\tau }^{k} \Pi _\mathcal {T}[\mathbf {u}_{h,\tau }^{k}], \mathbf {v}_{h,\tau }^{k} ) \cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} - \zeta \int _{\Omega _{h,\tau }} p(\varrho _{h,\tau }^{k}) \mathrm{div}{\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x}\nonumber \\&\qquad + \zeta \lambda \int _{\Omega _{h,\tau }} \mathrm{div}\mathbf {u}_{h,\tau }^{k} \mathrm{div}{\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} + \zeta 2\mu \int _{\Omega _{h,\tau }} \mathbf {D}( \mathbf {u}_{h,\tau }^{k} ) : \nabla {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} \nonumber \\&\qquad + (1-\zeta ) \mu \int _{\Omega _{h,\tau }} \frac{1}{\mathcal {F}^{k}} { \left( \nabla \mathbf {u}_{h,\tau }^k \mathbb {J}^k + (\mathbb {J}^k)^T \nabla ^T\mathbf {u}_{h,\tau }^k \right) : \left( \nabla {\varvec{\Psi }}_{h,\tau }\mathbb {J}^k \right) } \,\mathrm{d} {x}\nonumber \\&\qquad + \zeta 2\mu \sum _{\sigma \in \mathcal {E}_{\mathrm{I}}}\int _{\sigma } \frac{1}{h} \left[ \! \left[ \mathbf {u}_{h,\tau }^k\right] \! \right] \cdot \left[ \! \left[ {\varvec{\Psi }}_{h,\tau }\right] \! \right] \mathrm{d}S(x) \nonumber \\&\qquad + (1-\zeta ) 2\mu \sum _{\sigma \in \mathcal {E}_{\mathrm{I}}}\int _{\sigma } \frac{1}{h} \left[ \! \left[ \mathbf {u}_{h,\tau }^k\right] \! \right] \cdot \left[ \! \left[ {\varvec{\Psi }}_{h,\tau }\right] \! \right] \frac{1}{ | \mathcal {F}^{k}(\mathbb {J}^k)^{-T} \widehat{\mathbf {n}} | } \mathrm{d}S(x) ; \end{aligned}$$

(A.1b)

where ${\varvec{\Psi }}_{h,\tau }= ({\varvec{\Psi }}_{h,\tau }, {\psi }_{h,\tau }), \quad \eta _{h,\tau }^k = \eta _{h,\tau }^{k-1} + \tau z_{h,\tau }^k, \quad \mathbf {u}_{h,\tau }^k|_{\Sigma } = z_{h,\tau }^k \mathbf {e}_d , \quad \mathcal {F}_k^{k-1}= \left( H + \eta _{h,\tau }^{k-1}\right) /\left( H + \eta _{h,\tau }^{k}\right) .$

It is easy to check that F is continuous. Indeed, it is a one to one mapping, since the values of $\varrho ^\star$ and $U^\star$ can be determined by setting ${\varphi }_{h,\tau }= 1_{K}$ in (A.1a), and $(\Phi _\tau )_i=1_{D_\sigma }, (\Phi _\tau )_j=0$ for $j\ne i$ in (A.1b).

Let $(\varrho _{h,\tau }^k,U_{h,\tau }^k ) \in Q_h\times \mathbb {Q}$ and $\zeta \in [0,1]$ such that $F(\varrho _{h,\tau }^k,U_{h,\tau }^k , \zeta )=(0,0)$ (in particular $\varrho _{h,\tau }^k>0$). Then for any $\big ({\varphi }_{h,\tau }, {\varvec{\Phi }}_{h,\tau }= ({\varvec{\Psi }}_{h,\tau }, {\psi }_{h,\tau }) \big )\in Q_h\times \mathbb {Q}$

$$\begin{aligned}&\int _{\Omega _{h,\tau }} \frac{\varrho _{h,\tau }^k -\varrho _{h,\tau }^{k-1} \circ {X}_k^{k-1}\mathcal {F}_k^{k-1}}{\tau } {\varphi }_{h,\tau } \,\mathrm{d} {x} \nonumber \\&\qquad + \zeta \int _{\Omega _{h,\tau }} \mathrm{div}^{\mathrm{up}}_{h,\tau }( \varrho _{h,\tau }^{k}, \mathbf {v}_{h,\tau }^{k} ){\varphi }_{h,\tau } \,\mathrm{d} {x} =0; \end{aligned}$$

(A.2a)

$$\begin{aligned}&\int _{\Omega _{h,\tau }} \frac{\varrho _{h,\tau }^k \Pi _\mathcal {T}[\mathbf {u}_{h,\tau }^{k}] -(\varrho _{h,\tau }^{k-1}\Pi _\mathcal {T}[\mathbf {u}_{h,\tau }^{k-1}]) \circ {X}_k^{k-1}\mathcal {F}_k^{k-1}}{\tau } \cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} \nonumber \\&\qquad + \int _{\Sigma } \frac{ z_{h,\tau }^{k} - z_{h,\tau }^{k-1} }{\tau }{\psi }_{h,\tau } \mathrm{d}r\nonumber \\&\qquad + \int _{\Sigma } \Delta \eta _{h,\tau }^{k} \Delta {\psi }_{h,\tau } \mathrm{d}r - \int _{\Omega _{h,\tau }} \varrho _{h,\tau }^k \mathbf {f}_\tau ^{k}\cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} + \int _{\Sigma } g_\tau ^{k} {\psi }_{h,\tau } \mathrm{d}r\nonumber \\&\qquad +\zeta \int _{\Omega _{h,\tau }} \mathrm{div}^{\mathrm{up}}_{h,\tau }(\varrho _{h,\tau }^{k} \Pi _\mathcal {T}[\mathbf {u}_{h,\tau }^{k}], \mathbf {v}_{h,\tau }^{k} ) \cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} \nonumber \\&\qquad -\zeta \int _{\Omega _{h,\tau }} p(\varrho _{h,\tau }^{k}) \mathrm{div}{\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} + \zeta \lambda \int _{\Omega _{h,\tau }} \mathrm{div}\mathbf {u}_{h,\tau }^{k} \mathrm{div}{\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x}\nonumber \\&\qquad + \zeta 2\mu \int _{\Omega _{h,\tau }} \mathbf {D}( \mathbf {u}_{h,\tau }^{k} ) : \nabla {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} \nonumber \\&\qquad +(1-\zeta ) \mu \int _{\Omega _{h,\tau }} \frac{1}{\mathcal {F}^{k}} { \left( \nabla \mathbf {u}_{h,\tau }^k \mathbb {J}^k + (\mathbb {J}^k)^T \nabla ^T\mathbf {u}_{h,\tau }^k \right) : \left( \nabla {\varvec{\Psi }}_{h,\tau }\mathbb {J}^k \right) } \,\mathrm{d} {x}\nonumber \\&\qquad + \zeta 2\mu \sum _{\sigma \in \mathcal {E}_{\mathrm{I}}}\int _{\sigma } \frac{1}{h} \left[ \! \left[ \mathbf {u}_{h,\tau }^k\right] \! \right] \cdot \left[ \! \left[ {\varvec{\Psi }}_{h,\tau }\right] \! \right] \mathrm{d}S(x) \nonumber \\&\qquad + (1-\zeta ) 2\mu \sum _{\sigma \in \mathcal {E}_{\mathrm{I}}}\int _{\sigma } \frac{1}{h} \left[ \! \left[ \mathbf {u}_{h,\tau }^k\right] \! \right] \cdot \left[ \! \left[ {\varvec{\Psi }}_{h,\tau }\right] \! \right] \frac{1}{ | \mathcal {F}^{k}(\mathbb {J}^k)^{-T} \widehat{\mathbf {n}} | } \mathrm{d}S(x) . \end{aligned}$$

(A.2b)

Taking ${\varphi }_{h,\tau }=1$ as a test function in (A.2a) we obtain

$$\begin{aligned} \left\Vert \varrho _{h,\tau }^k\right\Vert _{L^1(\Omega _{h,\tau })}= \int _{\Omega _{h,\tau }^k} \varrho _{h,\tau }^k \,\mathrm{d} {x} = \int _{\Omega _{h,\tau }^{k-1}} \varrho _{h,\tau }^{k-1} \,\mathrm{d} {x} >0, \end{aligned}$$

(A.3)

which indicates the boundedness of $\varrho _{h,\tau }^k$ in the $L^1$ norm, and thus in all norms as the problem is of finite dimension. Following the same argument as Lemma 3 we know that $\varrho _{h,\tau }^k \ge 0$ provided $\varrho _{h,\tau }^{k-1} \ge 0$.

Taking ${\varvec{\Phi }}_{h,\tau }=(\mathbf {u}_{h,\tau }^k, z_{h,\tau }^k)$ as the test function in (A.2b) and follow the proof of Theorem (1) gives

$$\begin{aligned} \left\Vert U_{h,\tau }^k \right\Vert := \left\Vert \nabla \mathbf {u}_{h,\tau }^k\right\Vert _{L^2(\Omega _{h,\tau })} + \left\Vert z_{h,\tau }^k\right\Vert _{L^2(\Sigma )} \le C_1 \end{aligned}$$

(A.4)

where $C_1$ depends on the data of the problem.

Further, let $K\in \mathcal {T}_{h,\tau }^{k}$ be such that $\varrho _K^k$ is the smallest, i.e., $\varrho _K^k \le \varrho _L^k$ for all $L\in \mathcal {T}_{h,\tau }^{k}$. We denote $K'= \mathcal {A}_{h,\tau }^{k-1}\circ (\mathcal {A}_{h,\tau }^{k})^{-1}(K)$. Then a straightforward computation gives

Thus $\varrho _{h,\tau }^k \ge \varrho _K^k \ge \frac{ | K' |}{ | K |} \frac{\varrho ^{k-1}_{K'} }{1 + \tau \zeta | (\mathrm{div}\mathbf {v}_{h,\tau }^k)_K | } >0.$ Consequently, by virtue of (A.4) $\varrho _{h,\tau }^k > \epsilon$, where $\epsilon$ depends only on the data of the problem. Further, we get from (A.3) that $\varrho _{h,\tau }^k \le \frac{ \int _{\Omega _{h,\tau }^{k-1}} \varrho _{h,\tau }^{k-1} \,\mathrm{d} {x}}{\min _{K\in \mathcal {T}_{h,\tau }^{k}} |K|}$, which indicates the existence of $C_2>0$ such that $\varrho _{h,\tau }^k <C_2$. Therefore, Hypothesis 1 of Theorem 6 is satisfied.

Next, we proceed to show that Hypothesis 2 of Theorem 6 is satisfied. Let $\zeta =0$ then the system $F(\varrho _{h,\tau }^k,U_{h,\tau }^k )=0$ reads

$$\begin{aligned}&\varrho _{h,\tau }^k = \varrho _{h,\tau }^{k-1} \circ {X}_k^{k-1}\mathcal {F}_k^{k-1}; \end{aligned}$$

(A.5a)

$$\begin{aligned}&\int _{\Omega _{h,\tau }} \frac{\varrho _{h,\tau }^k \Pi _\mathcal {T}[\mathbf {u}_{h,\tau }^{k}] -(\varrho _{h,\tau }^{k-1} \Pi _\mathcal {T}[\mathbf {u}_{h,\tau }^{k-1}] ) \circ {X}_k^{k-1}\mathcal {F}_k^{k-1}}{\tau } \cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x}\nonumber \\&\quad + \int _{\Sigma } \frac{ z_{h,\tau }^{k} - z_{h,\tau }^{k-1} }{\tau }{\psi }_{h,\tau } \mathrm{d}r\nonumber \\&\quad + \mu \int _{\Omega _{h,\tau }} \frac{1}{\mathcal {F}^{k}} { \left( \nabla \mathbf {u}_{h,\tau }^k \mathbb {J}^k + (\mathbb {J}^k)^T \nabla ^T\mathbf {u}_{h,\tau }^k \right) : \left( \nabla {\varvec{\Psi }}_{h,\tau }\mathbb {J}^k \right) } \,\mathrm{d} {x} \nonumber \\&\quad + 2\mu \sum _{\sigma \in \mathcal {E}_{\mathrm{I}}}\int _{\sigma } \frac{1}{h} \left[ \! \left[ \mathbf {u}_{h,\tau }^k\right] \! \right] \cdot \left[ \! \left[ {\varvec{\Psi }}_{h,\tau }\right] \! \right] \frac{1}{ | \mathcal {F}^{k}(\mathbb {J}^k)^{-T} \widehat{\mathbf {n}} | } \mathrm{d}S(x)\nonumber \\&\quad + \int _{\Sigma } \left( \alpha \Delta \eta _{h,\tau }^{k} \Delta {\psi }_{h,\tau }+ \beta \nabla \eta _{h,\tau }^{k} \nabla {\psi }_{h,\tau }\right) \mathrm{d}r \nonumber \\&\quad - \int _{\Omega _{h,\tau }} \varrho _{h,\tau }^k \mathbf {f}_{h,\tau }^{k}\cdot {\varvec{\Psi }}_{h,\tau } \,\mathrm{d} {x} + \int _{\Sigma } g_\tau ^{k} {\psi }_{h,\tau } \mathrm{d}r =0. \end{aligned}$$

(A.5b)

To solve the above system (A.5), we further reformulate it on the reference domain according to (2.10)

$$\begin{aligned}&\widehat{\varrho }_{h,\tau }^k \mathcal {F}^{k}= \widehat{\varrho }_{h,\tau }^{k-1}\mathcal {F}^{k-1}; \end{aligned}$$

(A.6a)

$$\begin{aligned}&\quad \int _{\widehat{\Omega }} \widehat{\varrho }_{h,\tau }^{k-1}\mathcal {F}^{k-1}\frac{\Pi _\mathcal {T}[\widehat{\mathbf {u}}_{h,\tau }^{k}] -\Pi _\mathcal {T}[\widehat{\mathbf {u}}_{h,\tau }^{k-1}]}{\tau } \cdot \widehat{\varvec{\Psi }}_{h,\tau } \,\mathrm{d} \widehat{x}\nonumber \\&\quad + \int _{\Sigma } \frac{ z_{h,\tau }^{k} - z_{h,\tau }^{k-1} }{\tau }{\psi }_{h,\tau } \mathrm{d}r + 2\mu \sum _{\sigma \in \widehat{\mathcal {E}_{\mathrm{I}}}}\int _{\sigma } \frac{1}{h} \left[ \! \left[ \widehat{\mathbf {u}}_{h,\tau }^k\right] \! \right] \cdot \left[ \! \left[ \widehat{\varvec{\Psi }}_{h,\tau }\right] \! \right] \mathrm{d}S(\widehat{x})\nonumber \\&\quad + 2\mu \int _{\widehat{\Omega }} \widehat{\mathbf {D}}( \widehat{\mathbf {u}}_{h,\tau }^{k} ) : \widehat{ \nabla } \widehat{\varvec{\Psi }}_{h,\tau } \,\mathrm{d} \widehat{x} + \int _{\Sigma } \left( \alpha \Delta \eta _{h,\tau }^{k} \Delta {\psi }_{h,\tau }+ \beta \nabla \eta _{h,\tau }^{k} \nabla {\psi }_{h,\tau }\right) \mathrm{d}r\nonumber \\&\quad - \int _{\widehat{\Omega }} \widehat{\mathbf {f}}_{\tau }^{k}\cdot \widehat{\varvec{\Psi }}_{h,\tau }\widehat{\varrho }_{h,\tau }^{k-1} \mathcal {F}^{k-1} \,\mathrm{d} \widehat{x} + \int _{\Sigma } g^{k} {\psi }_{h,\tau } \mathrm{d}r =0, \end{aligned}$$

(A.6b)

where $\mathcal {F}^{k-1}= 1+ \eta _{h,\tau }^{k-1}/H$ is determined by $\eta _{h,\tau }^{k-1}$. Realizing that (A.6b) is a linear system with a matrix being block-wise symmetric positive definite, we know that there exists exactly one solution $\widehat{U}_{h,\tau }^k =(\widehat{\mathbf {u}}_{h,\tau }^k, z_{h,\tau }^k)$. Then using the fact $\eta _{h,\tau }^k = \eta _{h,\tau }^{k-1} + \tau z_{h,\tau }^k$ we get $\eta _{h,\tau }^k$ and $\mathcal {A}_{h,\tau }^{k}$. Further, it is straightforward that $\mathbf {u}_{h,\tau }^k=\widehat{\mathbf {u}}_{h,\tau }^k\circ \mathcal {A}_{h,\tau }^{k}(\widehat{x})$. Finally, substituting $\eta _{h,\tau }^k$ into (A.6a) we obtain the solution for $\varrho _{h,\tau }^k$. Obviously, $\varrho _{h,\tau }^k>0$ as long as no self touching. Thus the solution $(\varrho _{h,\tau }^k,U_{h,\tau }^k )$ belongs to W, which implies Hypothesis 2 of Theorem 6.

We have shown that both hypotheses of Theorem 6 hold. Applying Theorem 6 finishes the proof. $\square$

1.2 A.2 Proof of Lemma 8: renormalization

Here we show the validity of the discrete renormalized equation stated in Lemma 8 for the discrete continuity problem (4.17a).

Proof

Firstly, we set ${\varphi }_{h,\tau }=B'(\varrho )$ in (4.17a) and obtain

Next, recalling (3.4), we know there exist $\xi \in \mathrm{co}\{ \varrho _{h,\tau }^{k-1}\circ {X}_k^{k-1} , \varrho _{h,\tau }^{k} \}$ such that

where

Further, by recalling the definition of the upwind flux (4.10), and using again the Taylor expansion, we reformulate the convective term as

where

Moreover, using the facts

we obtain

Consequently, we derive

Finally, collecting the above terms and seeing $\mathbf {v}_{h,\tau }^k + \mathbf {w}_{h,\tau }^k= \mathbf {u}_{h,\tau }^k$, we complete the proof, i.e.,

$\square$

1.3 A.3 Proof of Theorem 4: energy stability

Here we prove the energy stability stated in Theorem 4 for the discrete scheme (4.17).

Proof

Setting ${\varphi }_{h,\tau }= - \frac{\left| \Pi _\mathcal {T}[ \mathbf {u}_{h,\tau }^{k}]\right| ^2}{2}$ in (4.17a) and $({\varvec{\Psi }}_{h,\tau },{\psi }_{h,\tau }) = (\mathbf {u}_{h,\tau }^{k}, z_{h,\tau }^{k})$ in (4.17b) we get $\sum _{i=1}^2 I_i=0$ and $\sum _{i=3}^{9} I_i=0$ respectively, where

Now we proceed with the summation of all the $I_i$ terms for $i=1,\ldots ,9$.

Term $(I_1+I_3+I_8)+(I_6+I_7)+I_9$. Firstly, analogously as in the proof of Theorem 1 we have

Term $I_2+I_4$. For the convective terms, we have using the fact that $\Pi _\mathcal {T}[\mathbf {u}_{h,\tau }]$ and $\mathrm{div}^{\mathrm{up}}_{h,\tau }\left( \varrho _{h,\tau }^k \Pi _\mathcal {T}[ \mathbf {u}]^{k}_h , \mathbf {v}_{h,\tau }^{k} \right)$ are constant on each $K\in \mathcal {T}_{h,\tau }$ and the upwind divergence

Pressure term $I_{5}$. Recalling the discrete internal energy equation (4.20), we can rewrite the pressure term as

where $D_1$ and $D_2$ are given in (4.21). Collecting all the above terms, we get

We finish the proof by summing up the above equation for $k=1,\ldots ,N$ and multiplying with $\tau$. $\square$

1.4 A.4 Proof of Lemma 10: useful estimates

Proof

Item 1 has been reported by [23, Lemma 3.5]. Item 2 has been reported by [31, Lemma 4.3]. Item 4 has been reported by [25, Chaper 9, Lemma 7]. We are only left with the proof of Item 3. We start the proof with the a-priori estimates on $\mathbf {v}_{h,\tau }$

where we used (4.14) for $\mathbf {u}_{h,\tau }$ and (4.28) for $\mathbf {w}_{h,\tau }$. On one hand, for $\gamma \ge 2$, we employ (4.30) to get

On the other hand, it is easy to check for $\gamma \in (1,2)$ that $\mathcal {H}''(r)=a r^{\gamma -2} \ge a$ if $r\le 1$ and $r \mathcal {H}''(r)= ar^{\gamma -1} \ge a$ if $r\ge 1$. Therefore

$$\begin{aligned} \mathcal {H}''(r)(1+r) \ge a \text{ for } \text{ all } r \in (0,\infty ) \end{aligned}$$

Applying these inequalities together with Hölder’s inequality, and the estimate (4.22) we derive (by choosing $\varrho _{h,\tau }^\dagger$ conveniently and (4.13)) that

Then, for $\gamma \in [6/5, 2)$ we have $I_1 {\mathop {\sim }\limits ^{<}}h^{-\frac{1}{2}}\tau ^{-\frac{1}{4}}$. Concerning $\gamma \in (1,6/5)$ we deduce by inverse estimate (4.14) that

which completes the proof of the first estimate (4.31a).

Similarly, we prove the second estimate (4.31b) in two steps. First for $\gamma \ge 2$ we may derive it due to Hölder’s inequality, trace theorem, and the inverse estimate (4.14) that

Next, we proceed to show the second estimates for $\gamma \in (1, 2)$.

where we have used the algebraic inequality for $\gamma \in (1,2)$ that . On the other hand, if $1<\gamma < \frac{4}{3}$ we complete the proof by the inverse estimates (4.14) and find

which finishes the estimate. $\square$