## Section: Scientific Foundations

### Approximation methods

All the mathematical models considered and studied in Smash consist in hyperbolic systems of PDE's. Most of the attention is focused on the 7 equation model for non-equilibrium mixtures and the 5 equation model for mechanical equilibrium mixtures. The main difficulty with these models is that they cannot be written under divergence form. Obviously, the conservation principles and the entropy inequality are fulfilled, but some equations (the volume fraction equation in particular) cannot be cast under conservative form. From a theoretical point of view, it is known since the works of Schwartz [53] that the product of two distributions is not defined. Therefore, the question of giving a sense to this product arises and as a consequence, the numerical approximation of non-conservative terms is unclear [28] , [39] . Aware of this difficulty, we have developed two specific methods to solve such systems.

The first one is the *discrete equations method* (DEM) presented
previously as a new homogenization method. It is moreover a numerical
method that solves non-conservative products for the 7 equation model
in the presence of shocks. With this method, Riemann problem solutions
are averaged in each sub-volume corresponding to the phase volumes in
a given computational cell. When a shock propagates inside a cell,
each interaction with an interface, corresponds to the
location where non-conservative products are undefined. However, at
each interaction, a diffraction process appears. The shock
discontinuity splits in several waves : a left facing reflected wave,
a right facing transmitted wave and a contact wave. The interface
position now corresponds to the one of the contact wave. Along its
trajectory, the velocity and pressure are now continuous : this is a
direct consequence of the diffraction process. The non-conservative
products that appear in these equations are precisely those that
involve velocity, pressure and characteristic function gradient. The
characteristic function gradient remains discontinuous at each
interface (it corresponds to the normal) but the other variables are
now continuous. Corresponding non-conservative products are
consequently perfectly defined : they correspond to the local
solution of the Riemann problem with an incoming shock as initial
data. This method has been successfully developed and validated in
many applications
[1] , [7] , [5] ,
[27] .

The second numerical method deals with the numerical approximation of
the *five equation model* . Thanks to the shock relations previously
determined, there is no difficulty to solve the Riemann
problem. However, the next step is to average (or to project) the
solution on the computational cell. Such a projection is not trivial
when dealing with a non-conservative variable. For example, it is
well known that pressure or temperature volume average has no
physical meaning. The same remark holds for the *cell average* of
volume fraction and internal energy. To circumvent this difficulty
a new relaxation method has been built
[16] . This method uses *two main
ideas* .

The first one is to *transform* one of the *non-conservative
products* into a *relaxation term* . This is possible with the
volume fraction equation, where the non conservative term corresponds
to the asymptotic limit of a pressure relaxation term. Then, a
splitting method is used to solve the corresponding volume
fraction equation. During the hyperbolic step, there is no difficulty
to derive a positivity preserving transport scheme. During the stiff
relaxation step, following preceding analysis of pressure relaxation
solvers [3] , there is no difficulty neither
to derive entropy preserving and positive relaxation solvers.

The second idea deals with the *management of the phase's energy
equations* , which are also present under *non-conservative
form* . These equations are able to compute regular/smooth solutions,
such as expansion waves, but are inaccurate for shocks. Thus they are
only used at shocks to predict the solution. With the predicted internal
energies, phase's pressures are computed and then *relaxed to
equilibrium* . It results in an *approximation* of the volume
fraction at shocks. This approximation is then used in the *mixture
equation of state* , that is unambiguously determined. This equation
of state is based on the *mixture energy* , a supplementary
equation. This equation, apparently redundant, has to be fulfiled
however. Its numerical approximation is obvious even in the presence
of shocks since it is a conservation law. With the help of the mixture
energy and predicted volume fraction, the *mixture pressure* is
now computed, therefore closing the system. This treatment guarantees
*correct* , *convergent* and *conservative wave
transmission* across material interfaces separating pure media. When
the interface separates a fluid and a mixture of materials, the
correct partition of energies among phases is fulfiled by replacing at
the shock front the internal energy equations by their corresponding
jumps [52] . To ensure the numerical solution strictly
follows the phase's Hugoniot curves, the poles of these curves are
transported [13] . With this
treatment, the method also converges for multiphase shocks.

This method is very *efficient* and *simple to
implement* . This also helped us considerably to solve very large
systems of hyperbolic equations, like those arising for elastic
materials in large deformations. The fluid-solid coupling via diffuse
interfaces with extreme density ratios was done efficiently, as shown
in Figure 5 .

Another difficulty encountered in solving two-phase flow problems comes from the high disparity between the wave speeds of each existing fluid material. In particular, one of the fluids may be very close to the incompressibility limit. In that case, we face up the problem of very low Mach number flows. The numerical treatment of these flows is still a problem and involves non trivial modifications of the original upwind schemes [38] , [37] . Our investigations in that domain concern both acoustic and incompressible aspects in methodologies for setting up suitable numerical methods.