Abstract
Abstract. We recall two conservative schemes recently proposed for the numerical solution of hyperbolic interface problems. Then, we compare the two schemes on a piston problem and a shock tube problem.
Introduction
We compare two conservative schemes for the system of conservation laws describing two gases separated by an interface. For the comparison we use a model problem introduced by Fazio and LeVeque [2] . A material interface I(t) separates two gases within a tube with a piston L(t) at one of its ends. The governing equations for this problem are given by Euler equations of gas dynamics,
Introduction
with q = [ρ, ρu, E] T , f (q) = ρu, ρu 2 + p, (E + p)u T ,
Introduction
and with the constitutive law for ideal gases
Introduction
where ρ, u, E, and p denote, respectively, density, velocity, total energy density per unit volume, and pressure of the gas. The polytropic constant γ(x, t) takes the value γ 1 for 0 ≤ x < I(t), and γ 2 for I(t) < x ≤ L(t).
Introduction
The motion of the piston is governed by Newton's equation
Introduction
where A is the area of the piston, m is its mass and p out (t) is a prescribed external pressure. From the above model a tube problem is recovered by assuming that the piston is initially at rest and setting A = 0 in (4).
The Eulerian scheme
For the Eulerian scheme we apply a finite volume approach, where we denote by
The Eulerian scheme
the overage of the filed quantities on the reference grid-cell. Within the physical domain we define a moving mesh, x n i < x n i+1 at time t n with ∆x n i = x n i+1 − x n i and ∆t n = t n+1 − t n . Moreover, we assume a constant speed of the moving mesh, that iṡ
The Eulerian scheme
Integrating the conservation law over the trapezoid (shown shaded in figure 1 ), we get Let us define κ n i = ∆x n i /∆η, where κ(η, t) is a capacity function and η is a computational variable with constant ∆η. Then, replacing ∆x n i by κ n i ∆η in the above equation and dividing by ∆η, we get
The Eulerian scheme
This gives a consistent, but nonconservative, approximation to the balance law because
The Eulerian scheme
Note that, when the grid is moving, κ n i varies with n. The algorithm should achieves two goals:
The Eulerian scheme
should be conserved with n;
The Eulerian scheme
2. constant states Q should be preserved, even when κQ is varying (i.e., when the mesh is moving).
The Eulerian scheme
To this end, we can use the relation
The Eulerian scheme
The above equation, after some rearrangements, gives
The Eulerian scheme
This can be viewed as a conservative discretization of the balance law (see [2] ).
The Eulerian scheme
This moving-mesh method supplemented by suitable monitor functions has been used by J. M. Stockie, J. A. Mackenzie and R. D. Russell [6] to provide a better resolution of wave structures, in particular of contact discontinuities, in comparison with fixed mesh computations.
Stability analysis
As far as the stability of the moving-mesh method is concerned, by following Harten and Hyman [4] , we require that the domain of influence of the Riemann problem at
Stability analysis
. Hence, we impose that
Stability analysis
where 
Stability analysis
are shifted wave speeds. The two above stability restrictions can be unified into
Stability analysis
Provided thatẋ n i ≈ λ p i we get wider time steps ∆t n with respect to a fixed nonuniform grid. This can be seen as a possibility to relax the global CFL condition. However, let us note that we have three waves and only one mesh speed, so that, in general we have to find a compromise.
Stability analysis
To grasp the reason for the shifted wave speeds in the stability condition (5), we write the transformed system
Stability analysis
in the computational variable η (here subscripts stand for partial derivatives), and notice that the new Jacobian of the flux functions is given by the old one minus the mesh speed.
Stability analysis
A different stability condition follows by requiring ∆t n to be small enough so that waves from neighboring cells do not interact. This results in a more strict stability condition, where the left-hand side is one half than the one in (5). However, numerical tests confirm that stability is ensured by imposing (5).
The Lagrangian scheme
In this section, we follow the treatment by Fazio and Russo [3] .
Lagrangian formulation
By introducing the Lagrangian coordinate ξ given by
Lagrangian formulation
where x 0 (t) denotes the Eulerian coordinate of the first fluid particle of the domain, the Euler equations (1)-(2) can be transformed in Lagrangian form
Lagrangian formulation
which is in conservation form too, with
Lagrangian formulation
here the time derivative is the Lagrangian derivative
Lagrangian formulation
the new field variables are defined by V (ξ, t) = ρ −1 , E = E/ρ, and the equation of state (3) becomes
Lagrangian formulation
The inverse transformation of coordinate is
Lagrangian formulation
and x 0 (t) satisfies the equation
Lagrangian formulation
Hence, 0 ≤ ξ ≤ ξ max will be our "computational domain" in which we have a fixed uniform grid with ξ i = (i − 1/2)∆ξ, for i = 1, 2, . . . , N , denoting the center of i-th cell, and ∆ξ = ξ max /N .
The Nessyahu and Tadmor central scheme.
The Nessyahu and Tadmor central scheme [5] has the form of a predictorcorrector scheme where ρ(A) denotes the spectral radius of the Jacobian matrix
The Nessyahu and Tadmor central scheme.
This condition ensures that the generalized Riemann problems with piecewise smooth data at time t n do not interact during the time step ∆t. q ′ j /∆ξ and f ′ j /∆ξ are first order approximations of the space derivatives of, respectively, the field and the flux, and can be computed in several ways. The simplest choice (used here) is
The Nessyahu and Tadmor central scheme.
where MM(v, w) is the min-mod limiter
The Nessyahu and Tadmor central scheme.
Let us note that, after one time step, the numerical solution is computed on a staggered grid (see Figure 2) . After two time steps, the numerical solution is evaluated on the original grid. 
Balancing the pressure at the interface.
We note that the interface is a material line and, therefore, velocity and pressure have to be continuous across the interface. This subsection report a possible way to enforce the continuity of the pressure at the interface We assume that, at the initial time, the interface is located at the boundary between cell j w and cell j w + 1. Because of the use of Lagrangian coordinates, the interface will always separate cell j w from cell j w + 1 at even time steps, and it will be in the middle of cell j w + 1/2 at odd time steps (see Figure 3 ).
Balancing the pressure at the interface.
Let us denote by subscript 1 and 2 the values of the field variables on the two sides of the interface cell at odd time steps. The balance of pressure on Fig. 3 . Evolution of the specific volume V near the interface cell for a steady solution.
Balancing the pressure at the interface.
both sides of the interface, p 1 = p 2 , gives
Balancing the pressure at the interface.
We assume that, on the two sides of the interface, there are two different gases, whose ratio of densities, η, is equal to the ratio of molecular masses:
Balancing the pressure at the interface.
This condition is physically correct if the temperature is continuous across the interface. The cell average of specific volume V and energy E at the interface cell j w + 1/2 are
Balancing the pressure at the interface.
By using of above relations (10-11-12), we get the following formula for the pressure p = p 1 = p 2 in terms of the field variables at the interface
Balancing the pressure at the interface.
Let us note that, by setting η = 1 and γ 1 = γ 2 = γ, we recover from (13) the classical equation of state for a polytropic gas. Once the interface quantities are known, the values of the field variables on each of the two sides can be computed Figure 3 shows the evolution of specific volume obtained by our scheme on two time steps for a steady solution. A similar picture is obtained for the evolution of energy. Note how the scheme maintains static solutions.
Numerical tests
To validate the schemes we perform two tests. In particular, we consider the piston problem treated by Fazio and LeVeque [2] , and the shock-interface interaction problem by Abgrall and Karni [1] .
Numerical tests
For the first test, two gases are separated by an interface between a solid wall and a plane piston. The related initial and interface conditions are
Numerical tests
dI dt (t) = u 1 (I(t), t) = u 2 (I(t), t) ; We plot the evolution of the interface and the piston with time, in three cases, obtained with our two different methods. The agreement of the two approaches is remarkable, considering that only 100 grid points have been used. In the initial state the two gases are at rest, with constant pressure throughout the domain. The piston is set into motion by the (positive) difference between external and internal pressure. Let us observe that the interface does not move for a while. It starts moving when the acoustic wave, propagating inward from the piston, reaches it.
Numerical tests
The motion of the whole system is oscillatory. The period of oscillation for the case γ 1 = γ 2 = 1.4 is larger than the period for the case γ 1 = γ 2 = 2.8, and the case of two different gases γ 1 = 1.4 and γ 2 = 2.8 is intermediate between the other two. This represents a qualitative test because the above described behaviour is the correct physical one.
Numerical tests
The second test problem is the one proposed by Karni and Abgrall [1] . The related initial conditions are (see [1] , Test case 4).
Numerical tests
with γ 1 = 1.4 and γ 2 = 1.6. Figure 5 illustrates the numerical results obtained by the Eulerian scheme for the two fluid problem. In Figure 6 the calculation is repeated with the Lagrangian scheme. In both cases, we used 800 grid-points in order to permit a comparison of numerical results with the original ones obtained by Abgrall and Karni. The lack of resolution at the interface in Figure 6 is not due to our treatment of the interface, but rather to the relatively poor performace of central schemes (with no artificial compression) at contact discontinuities, when compared to upwind schemes.

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