$Y$ in $\\{\Gamma,\Pi\\}$, formally which may be viewed as relaxed maximum
principles. The limiters in (eq. 82) thus impose similar maximum principles:

$m_{\Gamma}\leq\tilde{\Gamma}^{0\leqslant k\leqslant p,n+1}_{j}\leqslant
M_{\Gamma},\quad m_{\Pi}\leqslant\tilde{\Pi}^{0\leqslant k\leqslant
p,n+1}_{j}\leqslant M_{\Pi}.$

##  8 Numerical experiments

We now perform numerical tests with the DGSEM for discretizing the SG-gamma
model (eq. 1)-(eq. 2) where the space discretization is defined in (eq. 29)
and where the three-stage third-order strong stability-preserving Runge-Kutta
scheme by Shu and Osher [57] is used for the time discretization. The time
step is evaluated from the CFL condition (eq. 80a) and the limiter, introduced
in section 7, is applied at the end of each stage. We consider numerical tests
from [14, 16, 15, 38, 17], and we refer to the work in [49] for additional
details on the numerical setups.

The DGSEM presented here was implemented in the CFD code Aghora developed at
ONERA [55]. We recall that here we have proposed two numerical schemes using
DGSEM for the SG-gamma model that differ in the fluctuations (eq. 30) in the
volume integral: a CP scheme with (eq. 44); a semi-discrete entropy stable
scheme with EC fluxes (eq. 45). We will compare the performance of both
schemes in this section. All numerical tests are performed at fourth-order
accuracy, $p=3$, in space unless stated otherwise.

###  8.1 Advection of a density wave

We begin by validating the high-order accuracy of the scheme (eq. 29) by
advecting a density wave in a uniform flow in the domain $\Omega=[0,1]^{2}$,
discretized with unstructured meshes with fourth-order curved elements (see
Figure 5(a)), with periodic conditions and initial condition
${\textbf{u}}_{0}({\textbf{x}})$:

$\alpha_{1_{0}}({\textbf{x}})=\frac{1}{2}+\frac{1}{4}\sin
4\pi(x+y),\quad\rho_{0}({\textbf{x}})=1+\frac{1}{2}\sin 2\pi(x+y),\quad
u_{0}({\textbf{x}})=v_{0}({\textbf{x}})=1,\quad\mathrm{p}_{0}({\textbf{x}})=1,$

along with the EOS parameters $\mathrm{C_{v}}_{1}=\mathrm{C_{v}}_{2}=1$,
$\gamma_{1}=1.4$, $\mathrm{p}_{\infty_{1}}=0$, $\gamma_{2}=3$,
$\mathrm{p}_{\infty_{2}}=2$. In this case, the density and EOS parameters are
purely convected in uniform velocity and pressure fields. Norms on the mixture
density error, $e_{h}=\rho_{h}-\rho$, under $h$\- and $p$-refinements are
displayed in Table 1 with either CP fluxes, or EC fluxes in the volume
integral. We observe that as the mesh is refined the expected $p+1$ order of
convergence is recovered by both schemes, but lower error levels are obtained
with the CP flux.

(a) (a)

(b) (b)

Figure 5: Examples of meshes used for the numerical tests: (a) square mesh $h=1/4$ with $16$ fourth-order curved quadrangles with representation of the nodes; (b) example of unstructured mesh in the range $2.5\leqslant x\leqslant 5$ with $1580$ elements and initial positions of the shock and bubble (see section 8.3). | $p$ | $h$ | $\left\lVert e_{h}\right\rVert_{L^{1}(\Omega_{h})}$ | $\mathcal{O}_{1}$ | $\left\lVert e_{h}\right\rVert_{L^{2}(\Omega_{h})}$ | $\mathcal{O}_{2}$ | $\left\lVert e_{h}\right\rVert_{L^{\infty}(\Omega_{h})}$ | $\mathcal{O}_{\infty}$
---|---|---|---|---|---|---|---|---
CP | 1 | 1/8 | 3.76E-01 | – | 4.16E-01 | – | 6.25E-01 | –
1/16 | 1.82E-01 | 1.05 | 2.04E-01 | 1.03 | 3.69E-01 | 0.76
1/32 | 5.47E-02 | 1.73 | 6.11E-02 | 1.74 | 1.08E-01 | 1.77
1/64 | 1.43E-02 | 1.93 | 1.59E-02 | 1.94 | 2.56E-02 | 2.08
2 | 1/8 | 1.17E-02 | – | 1.41E-02 | – | 3.32E-02 | –
1/16 | 1.06E-03 | 3.46 | 1.23E-03 | 3.52 | 3.64E-03 | 3.19
1/32 | 9.64E-05 | 3.45 | 1.17E-04 | 3.40 | 4.82E-04 | 2.90
1/64 | 1.01E-05 | 3.25 | 1.30E-05 | 3.16 | 6.50E-05 | 2.90
3 | 1/8 | 3.51E-04 | – | 4.68E-04 | – | 3.18E-03 | –
1/16 | 1.73E-05 | 4.35 | 2.65E-05 | 4.14 | 1.98E-04 | 4.01
1/32 | 1.05E-06 | 4.04 | 1.66E-06 | 4.00 | 1.20E-05 | 4.04
1/64 | 6.73E-08 | 3.96 | 1.05E-07 | 3.98 | 7.54E-07 | 3.99
EC | 1 | 1/8 | 3.72E-01 | – | 4.11E-01 | – | 6.24E-01 | –
1/16 | 1.78E-01 | 1.06 | 2.05E-01 | 1.01 | 4.29E-01 | 0.54
1/32 | 5.62E-02 | 1.66 | 7.14E-02 | 1.52 | 2.12E-01 | 1.02
1/64 | 1.52E-02 | 1.89 | 1.99E-02 | 1.84 | 6.81E-02 | 1.64
2 | 1/8 | 2.78E-02 | – | 3.48E-02 | – | 9.99E-02 | –
1/16 | 3.95E-03 | 2.81 | 5.23E-03 | 2.73 | 1.76E-02 | 2.50
1/32 | 3.56E-04 | 3.47 | 4.55E-04 | 3.52 | 1.56E-03 | 3.50
1/64 | 2.91E-05 | 3.61 | 3.77E-05 | 3.59 | 1.41E-04 | 3.46
3 | 1/8 | 4.58E-03 | – | 5.78E-03 | – | 1.62E-02 | –
1/16 | 2.35E-04 | 4.28 | 3.10E-04 | 4.22 | 1.10E-03 | 3.88
1/32 | 8.53E-06 | 4.78 | 1.21E-05 | 4.68 | 6.62E-05 | 4.06
1/64 | 2.91E-07 | 4.87 | 4.63E-07 | 4.71 | 4.01E-06 | 4.05
Table 1: Advection of a density wave: results using either CP (top) numerical
fluxes (eq. 44), or EC (bottom) numerical fluxes (eq. 45) in the volume
integral. Norms of the error on density under $p$\- and $h$-refinements and
associated orders of convergence at time $t=2$.

###  8.2 Riemann problems

We first consider the advection of an isolated material discontinuity to
assess the contact preservation property of the present scheme. The
computational domain for this test is chosen to be $\Omega_{h}=[-0.5,0.5]$
with $100$ elements. The Riemann initial data are

$(\alpha_{1},\rho,u,\mathrm{p})=\left\\{\begin{array}[]{ll}(0.375,2,1,1),&x<0,\\\
(0.146342,1,1,1),&x>0,\end{array}\right.$

with $\mathrm{C_{v}}_{1}=1$, $\mathrm{C_{v}}_{2}=2$, $\gamma_{1}=1.4$,
$\gamma_{2}=1.5$, $\mathrm{p}_{\infty_{1}}=\mathrm{p}_{\infty_{2}}=0$. Results
at time $t=0.2$ are displayed in Figure 6. As expected, computations with CP
fluxes preserve uniform profiles of pressure and velocity across the material
interface, while spurious oscillations occur when using EC fluxes. In both
cases, the interface is well captured with low amplitude oscillations in the
$\rho$ and $\Gamma$ profiles.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 6: Advection of an isolated material interface: fourth-order accurate
simulations obtained on a mesh with $100$ elements, and using either CP (top),
or EC (bottom) fluxes in the volume integral. Approximate results (symbols)
are shown at $t=0.2$ and are compared to the exact solution (lines).

We now consider a gas-gas shock-interface interaction problem which was
originally proposed in [44]. Here a shock wave in helium gas travels at Mach
$8.96$ and interacts with an helium-air interface. The computational domain
$\Omega_{h}=[-1,1]$ with $100$ elements and the initial condition is as
follows:

$(\alpha_{1},\rho,u,\mathrm{p})=\left\\{\begin{array}[]{ll}(0,0.386,26.59,100),&x<-0.8,\\\
(0,0.1,-0.5,1),&-0.8<x<-0.2,\\\ (1,1.0,-0.5,1),&x>-0.2,\end{array}\right.$

with $\mathrm{C_{v}}_{1}=1$, $\mathrm{C_{v}}_{2}=2.5$, $\gamma_{1}=1.4$,
$\gamma_{2}=5/3$, $\mathrm{p}_{\infty_{1}}=\mathrm{p}_{\infty_{2}}=0$. The
results are displayed in Figure 7, where the solution shows two shocks, one
traveling left and the other traveling right, with a right traveling material
interface in between. As a result, the shocks and interface are well captured,
while spurious oscillations of small amplitude occur in the pressure and
velocity fields.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 7: Shock-material interface interaction: fourth-order accurate, $p=3$,
simulations obtained on a mesh with $100$ elements, and using either CP (top),
or EC (bottom) fluxes in the volume integral. Approximate results (symbols)
are shown at $t=0.07$ and are compared to the exact solution (lines).

The last test case concerns a gas-water shock-interface interaction problem
and simulates an underwater explosion, where the initial condition consists of
a material interface separating highly compressed air to the left and water at
atmospheric pressure to the right. The computational domain is
$\Omega_{h}=[-5,5]$ with $100$ elements, and the initial data are given as

$(\alpha_{1},\rho,u,\mathrm{p})=\left\\{\begin{array}[]{ll}(1,1.241,0,2.753),&x<0,\\\
(0,0.991,0,3.059\times 10^{-4}),&x>0,\end{array}\right.$

with $\mathrm{C_{v}}_{1}=1.2$, $\mathrm{C_{v}}_{2}=0.073037$,
$\gamma_{1}=1.4$, $\gamma_{2}=5.5$, $\mathrm{p}_{\infty_{1}}=0$, and
$\mathrm{p}_{\infty_{2}}=1.505$. The results, in Figure 8, show a right
traveling shock, a right traveling contact wave, a right advected material
interface and a left rarefaction wave. We observe small oscillations on the
velocity and pressure profiles even though the shock is of large amplitude,
and the shock and material interface are well captured.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 8: Gas-water shock-interface interaction problem: fourth-order
accurate, $p=3$, simulations obtained on a mesh with $100$ elements, and using
either CP (top), or EC (bottom) fluxes in the volume integral. Approximate
results (symbols) are shown at $t=1$ and are compared to the exact solution
(lines).

###  8.3 Shock wave-helium bubble interaction

We now consider the interaction of a shock with a helium bubble, which was
experimentally investigated in [30] and used to assess numerical schemes for
multiphase and multicomponent flows [15, 29, 35, 38, 39, 54, 48]. The test
setup consists of a stationary helium bubble ($\gamma_{1}=1.648$ and
$\mathrm{C_{v}}_{1}=6.0598$) which is surrounded by air ($\gamma_{2}=1.4$ and
$\mathrm{C_{v}}_{2}=1.7857$) and interacts with a left moving Mach $1.22$
shock. The computational domain $\Omega_{h}=[0.0,6.5]\times[0,1.78]$ is
discretized with an unstructured mesh with $433016$ elements (see Figure
5(b)). The helium bubble of unit diameter is centered at $x=3.5$ and $y=0.89$
and the left traveling shock is located at $x=4$. Periodic boundary conditions
are imposed on the top and bottom boundaries, while non-reflective conditions
are applied at the left and right boundaries. The initial data are made
nondimensional with the initial bubble diameter, density, temperature and
sound speed of air in the pre-shock region.

We first test the ability of both schemes to preserve material interfaces and
consider the advection of the bubble only on a mesh with $64\times 64$ fourth-
order curved elements (see Figure 5(a)). We thus remove the shock wave, impose
a uniform velocity field ${\textbf{v}}_{0}({\textbf{x}})=(1,0)^{\top}$ and
reduce the size of the bubble which is now initially centered at $(0.5,0.5)$.
Figure 9 shows the bubble at time $t=76.19\mu s$ corresponding to a transport
of the bubble over a unit distance. The scheme with the CP fluxes captures the
interface sharply and preserves the uniform velocity and pressure profiles
across the interface, while spurious oscillations are observed with the EC
flux.

(a) (a) $t=0$

(b) (b) $t=76.19\mu s$, $y=0.5$

(c) (c) $t=76.19\mu s$, CP

(d) (d) $t=76.19\mu s$, EC

Figure 9: Advection of a helium bubble in air: (a) initial condition, and
results obtained with fourth-order accuracy $(p=3)$ in space with either CP
fluxes, or EC fluxes: (b) absolute error levels (logarithmic scale) on the
pressure and velocity distributions along $y=0.5$ obtained with either CP
(black symbols), or EC (red symbols) fluctuations; (c,d) numerical Schlieren
$\phi=\exp(|\nabla\rho|/|\nabla\rho|_{\max})$.

We now consider the shock-bubble interaction. Figure 10 shows the deformation
of the bubble at several physical times as the left traveling shock passes
through it and represents contours of void fraction of the helium bubble,
$\alpha_{1}$, and mixture pressure, $\mathrm{p}$, together with numerical
Schlieren. We observe that the scheme with CP fluxes allows a better and
sharper resolution of the bubble interface for all physical times and is able
to accurately capture the shock and bubble dynamics. The bubble interface
develops vortices after interacting with shock due to a Kelvin-Helmholtz
instability. Once again, results with the EC fluxes show some spurious
oscillations at the material interface before and after the interaction in
contrast to CP fluxes, while both CP and EC schemes show good resolutions of
the shock (shock is captured within, almost, one single cell).

$t=32\mu s$

(a)

(b)

(c)

(d)

(e)

$t=102\mu s$

(f)

(g)

(h)

(i)

(j)

$t=427\mu s$

(k)

(l)

(m)

(n)

(o)

$t=674\mu s$

(p)

(q) CP fluxes

(r) EC fluxes

(s) CP fluxes

(t) EC fluxes

Figure 10: Interaction of a $M=1.22$ shock in air with a helium bubble: fourth
order accurate in space, $p=3$, numerical simulations obtained at different
times using either CP, or EC fluxes in the volume integral, on an unstructured
mesh with $433016$ elements. For each snapshot the left image shows the helium
void fraction contours (color levels) and the pressure contours (lines), while
the right image shows the numerical Schlieren of the density
$\phi=\exp{|\nabla\rho|/|\nabla\rho_{\max}|}$.

###  8.4 Strong shock wave-hydrogen bubble interaction

We finally consider an interaction problem of a strong $M=2$ shock in air with
a hydrogen bubble that has been numerically investigated in [4, 59]. Compared
to section 8.3, these conditions result in faster shock and bubble dynamics.
The computational domain for this test is $\Omega_{h}=[0,22.5]\times[0,7.5]$
and is discretized with an unstructured mesh with $154622$ elements. The
hydrogen bubble ($\gamma_{1}=1.41$ and $\mathrm{C_{v}}_{1}=7.424$) is
initially centered at $x=4$ and $y=0$, and the right traveling shock located
at $x=7$ in air ($\gamma_{2}=1.353$ and $\mathrm{C_{v}}_{2}=0.523$). The
initial data is made nondimensional with the pre-shock density, velocity and
temperature of the air and a length scale of $1$mm. We impose symmetry
conditions at the top and bottom boundaries, along with supersonic inflow
condition at the left boundary and nonreflecting conditions at the right
boundary.

Figure 11 shows the deformation of the bubble as the shock passes through it,
where we plot contours of the void fraction of the hydrogen bubble,
$\alpha_{1}$, mixture pressure pressure, $\mathrm{p}$, and the numerical
Schlieren. Here, we once again observe that the numerical scheme is able to
resolve the bubble interface well along with the shock. The oscillations at
the interface are due to the Kelvin-Helmholtz instability and they were also
observed in [4]. Using CP fluxes maintains sharp resolution of the interface
while also proving to well capture shocks.

$t=1.5\mu s$

(a)

(b)

(c)

(d)

(e)

$t=2.5\mu s$

(f)

(g)

(h)

(i)

(j)

$t=4.0\mu s$

(k)

(l)

(m)

(n)

(o)

$t=8.8\mu s$

(p)

(q)

(r)

(s)

(t)

$t=13.6\mu s$

(u)

(v) CP fluxes

(w) EC fluxes

(x) CP fluxes

(y) EC fluxes

Figure 11: Interaction of a $M=2$ shock in air with a Hydrogen bubble: fourth-
order accurate in space, $p=3$, numerical solutions obtained at different
times using either CP, or EC numerical fluxes in the volume integral, on an
unstructured mesh with $154622$ elements. For each snapshot the left image
shows the helium void fraction contours (color levels) and the pressure
contours (lines), while the right image shows the numerical Schlieren of the
density $\phi=\exp{|\nabla\rho|/|\nabla\rho_{\max}|}$.

##  9 Concluding remarks

In this work, we propose a high-order, robust and entropy stable
discretization of the nonconservative multicomponent SG-gamma model [58]. The
space discretization of this system relies on the DGSEM framework [53, 15]
based on the modification of the integral over discretization elements where
we replace the physical fluxes and nonconservative products by two-point
numerical fluctuations. We first extend this framework to multidimensional
unstructured grids with curved elements and derive conditions under which the
semi-discrete scheme is high-order accurate, free-stream preserving and
entropy stable. We then design a robust, entropy stable, material interface
preserving, and maximum principle preserving HLLC solver for the SG-gamma
model that does not require a root-finding algorithm to evaluate the
nonconservative product. The HLLC solver is then used as interface
fluctuations in the proposed DG scheme, while we consider either EC, or CP
fluctuations in the integrals over discretization elements. The DGSEM is shown
to either satisfy a semi-discrete entropy inequality with EC fluctuations
(when excluding material interfaces), or to preserve material interfaces with
CP fluctuations.

We then analyze the fully discrete numerical scheme with a forward Euler time
discretization. We derive conditions on the time step to guarantee that the
cell-averaged solution remains in the set of states and satisfies a minimum
principle on the entropy and maximum principles on the EOS parameters with
either EC, or CP fluctuations. We use a posteriori scaling limiters [70, 71]
to extend these properties to all DOFs within elements, while a strong-
stability preserving Runge-Kutta scheme [57] is used for the high-order time
integration and keeps properties of the first-order in time scheme.

High-order accurate numerical simulations of flows in one and two space
dimensions with discontinuous solutions and complex wave interactions confirm
robustness, stability and accuracy of the present scheme with either EC, or CP
fluctuations, and the scheme with CP fluxes present better resolution
capabilities.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

This work was performed as part of P. Rai’s PhD thesis. The authors would like
to express their gratitude towards ONERA for providing the funds and resources
for this research work, and would like to thank the doctoral school at the
Ecole Polytechnique for their administrative support.

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