Remi Abgrall

A Simple Method for Compressible Multifluid Flows

A simple second order accurate and fully Eulerian numerical method is presented for the simulation of multifluid compressible flows, governed by the stiffened gas equation of state, in hydrodynamic regime. Our numerical method relies on a second order Godunov-type scheme, with approximate Riemann solver for the resolution of conservation equations, and a set of nonconservative equations. It is valid for all mesh points and allows the resolution of interfaces. This method works for an arbitrary number of interfaces, for breakup and coalescence. It allows very high density ratios (up to 1000). It is able to compute very strong shock waves (pressure ratio up to 10 5). Contrary to all existing schemes (which consider the interface as a discontinuity) the method considers the interface as a numerical diffusion zone as contact discontinuities are computed in compressible single phase flows, but the variables describing the mixture zone are computed consistently with the density, momentum and energy. Several test problems are presented in one, two, and three dimensions. This method allows, for example, the computation of the interaction of a shock wave propagating in a liquid with a gas cylinder, as well as Richtmeyer--Meshkov instabilities, or hypervelocity impact, with realistic initial conditions. We illustrate our method with the Rusanov flux. However, the same principle can be applied to a more general class of schemes.

A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems

We present a new Lagrangian cell-centered scheme for two-dimensional compressible flows. The primary variables in this new scheme are cell-centered, i.e., density, momentum, and total energy are defined by their mean values in the cells. The vertex velocities and the numerical fluxes through the cell interfaces are not computed independently, contrary to standard approaches, but are evaluated in a consistent manner due to an original solver located at the nodes. The main new feature of the algorithm is the introduction of four pressures on each edge, two for each node on each side of the edge. This extra degree of freedom allows us to construct a nodal solver which fulfills two properties. First, the conservation of momentum and total energy is ensured. Second, a semidiscrete entropy inequality is provided. In the case of a one-dimensional flow, the solver reduces to the classical Godunov acoustic solver: it can be considered as its two-dimensional generalization. Many numerical tests are presented. They are representative test cases for compressible flows and demonstrate the robustness and the accuracy of this new solver.

Discrete equations for physical and numerical compressible multiphase mixtures

We have recently proposed, in [21], a compressible two-phase unconditionally hyperbolic model able to deal with a wide range of applications: interfaces between compressible materials, shock waves in condensed multiphase mixtures, homogeneous two-phase flows (bubbly and droplet flows) and cavitation in liquids. One of the difficulties of the model, as always in this type of physical problems, was the occurrence of non-conservative products. In [21], we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces, not in the case of shock waves. This model was extended to several space dimensions in [24], In this paper, thanks to a deeper analysis of the model, we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities. This analysis provides a more accurate estimate of closure terms, but also an accurate resolution method for the conservative fluxes as well as non-conservative terms even for situations involving discontinuous solutions. The accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems.

Modelling phase transition in metastable liquids: application to cavitating and flashing flows

A hyperbolic two-phase flow model involving five partial differential equations is constructed for liquid-gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves in the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast at specific locations only, typically at evaporation fronts. Thus, metastable states are involved for locations far from these fronts. Computational results are compared to the experimental ones. Computed and measured front speeds are of the same order of magnitude and the same tendency of increasing front speed with initial temperature is reported. Moreover, the limit case of evaporation fronts propagating in highly metastable liquids with the Chapman-Jouguet speed is recovered as an expansion wave of the present model in the limit of stiff thermal and chemical relaxation.

High Order Fluctuation Schemes on Triangular Meshes

We develop a new class of schemes for the numerical solution of first-order steady conservation laws. The schemes are of the residual distribution, or fluctuation-splitting type. These schemes have mostly been developed in the context of triangular or tetrahedral elements whose degrees of freedom are their nodal values. We work here with more general elements that allow high-order accuracy. We introduce, for an arbitrary number of degrees of freedom, a simple mapping from a low-order monotone scheme to a monotone scheme that is as accurate as the degrees of freedom will allow. Proofs of consistency, convergence and accuracy are presented, and numerical examples from second, third and fourth-order schemes.

Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems

The aim of this paper is to construct upwind residual distribution schemes for the time accurate solution of hyperbolic conservation laws. To do so, we evaluate a space-time fluctuation based on a space-time approximation of the solution and develop new residual distribution schemes which are extensions of classical steady upwind residual distribution schemes. This method has been applied to the solution of scalar advection equation and to the solution of the compressible Euler equations both in two space dimensions. The first version of the scheme is shown to be, at least in its first order version, unconditionally energy stable and possibly conditionally monotonicity preserving. Using an idea of Csik et al. [Space-time residual distribution schemes for hyperbolic conservation laws, 15th AIAA Computational Fluid Dynamics Conference, Anahein, CA, USA, AIAA 2001-2617, June 2001], we modify the formulation to end up with a scheme that is unconditionally energy stable and unconditionally monotonicity preserving. Several numerical examples are shown to demonstrate the stability and accuracy of the method.

Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes

We present and analyze several ways of discretizing first-order Hamilton-Jacobi equations on unstructured meshes. We first discuss two Godunov-type Hamiltonians: the first one is an extension of a result by Bardi and Osher, where a particular decomposition of the initial condition is assumed, and we point out its practical limits; the other one arises from a particular decomposition of the Hamiltonian. Despite its complexity this decomposition enables us to construct a Lax-Friedrichs Hamiltonian. These schemes all share common properties: They are consistent, monotonic, and independent of the geometrical interpretation of the piecewise linear initial condition. Under these assumptions and classical ones on the mesh, we show these schemes are convergent. We describe their high-order extensions using the ENO technique and provide numerical illustrations.

Short Note: A comment on the computation of non-conservative products

We are interested in the solution of non-conservative hyperbolic systems, and consider in particular the so-called path-conservative schemes (see e.g. [2,3]) which rely on the theoretical work in [1]. The example of the standard Euler equations for a perfect gas is used to illuminate some computational issues and shortcomings of this approach.

Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems

The Residual Distribution (RD) schemes are an alternative to standard high order accurate finite volume schemes. They have several advantages: a better accuracy, a much more compact stencil, easy parallelization. However, they face several problems, at least for steady problems which are the only cases considered here. The solution is obtained via an iterative method. The iterative convergence must be good in order to get spatially accurate solutions, as suggested by the few theoretical results available for the RD schemes. In many cases, especially for systems, the iterative convergence is not sufficient to guaranty the theoretical accuracy. In fact, up to our knowledge, the iterative convergence is correct in only two cases: for first order monotone schemes and the (scalar) Struijs PSI scheme which is a multidimensional upwind scheme. Up to our knowledge, the iterative convergence is poor for systems, except for the blended scheme of Deconinck et al. [A. Csik, M. Ricchiuto, H. Deconinck, A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws, J. Comput. Phys. 179(2) (2002) 286-312] and Abgrall [R. Abgrall, Toward the ultimate conservative scheme: following the quest, J. Comput. Phys. 167(2) (2001) 277-315] which are also a genuinely multidimensional upwind scheme. A second drawback is that their construction relies, up to now, on a single first order scheme: the N scheme. However, it is known that standard first order finite volume schemes can be rephrased into a Residual Distribution framework. Unfortunately, the standard way of upgrading the order of accuracy to second order leads to very unsatisfactory results but clearly the construction of good schemes based on a wider class of first order schemes would be interesting. In this paper, we analyze these two problems, and show they are linked. We propose a fix and demonstrate its efficiency on several test cases that cover a wide range of applications. Our solution extends considerably the number of working RD schemes.

Two-Layer Shallow Water System: A Relaxation Approach

The two-layer shallow water system is an averaged flow model. It forms a nonconservative system which is only conditionally hyperbolic. The coupling between the layers, due to the hydrostatic pressure assumption, does not provide explicit access to the system eigenstructure, which is inconvenient for Riemann solution based numerical schemes. We consider a relaxation approach which offers greater decoupling and accessible eigenstructure. The stability of the model is discussed. Numerical results are shown for unsteady flows as well as for smooth and nonsmooth steady flows.