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In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the {restricted} case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions., 29 pages

Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation limit which can be solved numerically much more efficiently. For the Jin–Xin relaxation system in such a two-scale setting, we establish its wellposedness and singular limit as the (smaller) relaxation time goes to zero. The limit is a multiscale coupling problem which couples the original Jin–Xin system on the domain when the relaxation time is O(1) with its relaxation limit in the other domain through interface conditions which can be derived by matched interface layer analysis.As a result, we also establish the well-posedness and regularity (such as boundedness in sup norm with bounded total variation and L1-contraction) of the coupling problem, thus providing a rigorous mathematical foundation, in the general nonlinear setting, to the multiscale domain decomposition method for this two-scale problem originally proposed in Jin et al. in Math. Comp. 82, 749–779, 2013.

We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is based on a piecewise constant discretization of the cross-section and on an approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the use of a relaxation approximation that leads to a positive and entropy satisfying numerical scheme for all variation of section, even discontinuous sections with arbitrary large jumps. To do so, we introduce, in the first step of the relaxation solver, a singular dissipation measure superposed on the standing wave, which enables us to control the approximate speeds of sound and thus the time step, even for extreme initial data.

International audience; We study a class of models of compressible two-phase flows. This class, which includes the Baer-Nunziato model, is based on the assumption that each phase is described by its own pressure, velocity and temperature and on the use of void fractions obtained from averaging process. These models are nonconservative and non-strictly hyperbolic. We prove that the mixture entropy is non-strictly convex and that the system admits a symmetric form.

International audience; A family of explicit adaptive algorithms is designed to solve nonlinear scalar one-dimensional conservation laws. Based on the Godunov scheme on a uniform grid, a first strategy uses the multiresolution analysis of the solution to design an adaptive grid that evolves in time according to the time-dependent local smoothness. The method is furthermore enhanced by a local time-stepping strategy. Both numerical schemes are shown to converge towards the unique entropy solution.

For a class of non-conservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial-value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to non-conservative systems of a similar concept introduced by Abeyaratne, Knowles and Truskinovsky for subsonic phase transitions and by LeFloch for non-classical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for non-conservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase-plane analysis of travelling-wave solutions associated with an augmented version of the non-conservative system. We illustrate with several examples that non-conservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of travelling waves which yields the corresponding kinetic function.

An e cient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving unsteady second-order partial di erential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The implicit procedure uses a fast iterative algorithm with reduced evaluation cost introduced in [Renac, Marmignon and Coquel, to appear in SIAM J. Sci. Comput.]. The method is here extended to convection dominated ow problems. A second-order discretization in time is achieved by decomposing the integrations of convective and di usive terms with a splitting method. Numerical examples are presented for the linear convection-di usion equation in one and two space dimensions. The performance of the present method is seen to be improved in terms of CPU time when compared to a full implicit discretization of the parabolic terms in a wide range of Peclet numbers.

Well-balanced or asymptotic preserving schemes are receiving an increasing amount of interest. This paper gives a precise setting for studying both properties in the case of Euler system with friction. We derive a simple solver which, by construction, preserves discrete equilibria and reproduces at the discrete level the same asymptotic behavior as that of the solutions of the continuous system. Numerical illustrations are convincing and show that not all methods share these properties.

This paper is concerned with the numerical approximation of the solutions of a two-fluid two-pressure model used in the modelling of two-phase flows. We present a relaxation strategy for easily dealing with both the nonlinearities associated with the pressure laws and the nonconservative terms that are inherently present in the set of convective equations and that couple the two phases. In particular, the proposed approximate Riemann solver is given by explicit formulas, preserves the natural phase space, and exactly captures the coupling waves between the two phases. Numerical evidences are given to corroborate the validity of our approach.

The semilinear relaxation was introduced by Jin and Xin [Comm. Pure Appl. Math.48, 235, (1995)] in order to approximate the conservation law ∂tu + ∂xf(u) = 0 for any flux function f ∈ 𝒞1 (ℝ;ℝ). In this paper, we propose an alternative relaxation technique for scalar conservation laws of the form ∂tu + ∂xu(1 - u)g(u) = 0, where g ∈ 𝒞1 ([0, 1]; ℝ) and 0 ∉ g(]0, 1[). We extend this new philosophy to an arbitrary flux function f whenever possible. Unlike the semilinear approach, the new relaxation strategy does not involve any tuning parameter, but makes use of the Born–Infeld system. Another advantage of this method is that it enables us to achieve a maximum principle on the velocities w = (1 - u)g and z = -ug, which turns out to be a physically interesting and helpful feature in the context of some two-phase flow problems.

The MAMCDP'09 workshop took place in Paris in January 2009. It was intended to promote multiresolution and other adaptive techniques for complex applications where convection is the prevailing phenomenon. Presentations were given by both senior and young researchers from various institutions over two days. In this introduction we summarize the presentations whose slides are available on the website of the workshop as well as the present contributions.

Resume. Cet article aborde la question de l'utilisation efficace des maillages adaptatifs en vue de simuler correctement le transport des fronts sur des temps longs avec un cout CPU raisonnable. Ces singularites proviennent des systemes de lois de conservation modelisant lesdiphasiques ` travers les conduites petrolieres. Nous proposons non seulement un cadre multiresolution afin d'ajuster le pas d'espacela regularite locale de la solution, mais aussi une strategie de pas de temps local afin d'alleger les contraintes duesla condition CFL de stabilite. Nous examinerons plus particulierement le choix optimal des micro pas de tempsl'interieur d'un macro pas de temps. Nous presentons les simulations numeriques pour illustrer l'efficacite des methodes preconisees. Abstract. This paper is concerned with how to make effective use of adaptive grids to correctly simulate the transportation of fronts over a long range of time under reasonable CPU time. These step gradients arise from systems of conservation laws modelling two-phase flows in pipelines. We propose not only a multi-resolution setup in order to adapt the space grid to the local smoothness of the solution, but also a local time stepping strategy in order to alleviate the constraints due to the CFL stability condition. A special focus is given to the optimal choice of micro time steps within a macro time step. Numerical simulations are presented to illustrate the efficiency of the methods put forward.

The present work is devoted to an extension of the Navier–Stokes equations where the fluid is governed by two independent pressure laws. Several turbulence models typically enter this framework. The striking novelty over the usual Navier–Stokes equations stems from the impossibility to recast equivalently the system of interest in full conservation form. Opposing to systems of conservation laws, where the end states of the viscous shock are completely characterized by jump relations, the lack of conservation implies the absence of jump relations. We analyze the traveling wave behaviors according to the ratio of viscosities, and we show that the traveling wave solutions of our system tend to the traveling wave solutions of a fully conservative system. This result is used to exhibit asymptotic expansions of the end states. Such an asymptotic behavior achieves a deep physical interpretation when illustrated in the case of compressible turbulent flows.

We study the asymptotic behavior of the solutions of barotropic two-phase two-pressure models, with pressure relaxation, drag force and external forces. Using Chapman-Enskog expansions close to the expected equilibrium, a drift-flux model with a Darcy type closure law is obtained. Also, restricting this closure law to permanent flows (defined as steady flows in some Lagrangian frame), we can obtain a drift-flux model with an algebraic closure law, in the spirit of Zuber-Findlay models. The example of a two-phase flow in a vertical pipe is described.

SUMMARY This paper studies a global relaxation method to ensure the conservative coupling at a fixed interface of two Euler systems with different pressure laws. Copyright q 2007 John Wiley & Sons, Ltd.

International audience; We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer-Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in [16] for the isentropic Baer-Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer-Nunziato model, namely Schwendeman-Wahle-Kapila's Godunov-type scheme [39] and Toro-Tokareva's HLLC scheme [42]. The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.

International audience; We propose a dynamic model adaptation method for a nonlinear conservation law coupled with an ordinary differential equation. This model, called the ''fine model", involves a small time scale and setting this time scale to 0 leads to a classical conservation law, called the ''coarse model", with a flux which depends on the unknown and on space and time. The dynamic model adaptation consists in detecting the regions where the fine model can be replaced by the coarse one in an automatic way, without deteriorating the accuracy of the result. To do so, we provide an error estimate between the solution of the fine model and the solution of the adaptive method, enabling a sharp control of the different parameters. This estimate rests upon stability results for conservation laws with respect to the flux function. Numerical results are presented at the end and show that our estimate is optimal.

International audience; This paper is devoted to the mathematical and numerical analysis of a coupling procedure for one-dimensional Euler systems. The two systems have different closure laws and are coupled through a thin fixed interface. Following the work of [5], we propose to couple these systems by a bounded vector-valued Dirac measure, concentrated at the coupling interface, which in the applications may have a physical meaning. We show that the proposed framework allows to control the coupling conditions and we propose an approximate Riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem. Numerical experiments in constrained optimization problems are then presented to assess the performances of the present method. 1. Introduction The study of large-scale and complex problems exhibiting a wide range of physical space and time scales (see for instance [62, 35, 14]), usually requires separate solvers adapted to the resolution of specific scales. This is the case of many industrial flows. Let us quote, for example, the numerical simulation of two-phase flows applied to the burning liquid oxygen-hydrogen gas in rocket engines [58]. This kind of flow contains both separated and dispersed two-phase flows, due to atomization and evaporation phenomena. This requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled. Another example concerns turbomachine flows which can be modeled by the Euler equations of gas dynamics with different closure laws between the stages of the turbine, where the conditions of temperature and pressure are strongly heterogeneous. The coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine. The method of interface coupling allows to represent the evolution of such flows, where different models are separated by fixed interfaces. First, coupling conditions are specified at the interface to exchange information between the systems. The definition of transmission conditions generally results from physical consideration, e.g. the conservation or the continuity of given variables. Then, the transmission conditions are represented at the discrete level. The study of interface coupling for nonlinear hyperbolic systems has received attention for several years. In [43], the authors study the scalar case from both mathematical and numerical points of view.

We wish to demonstrate that it is judicious to combine various existing computational techniques that appeared for academic cases in seemingly unrelated areas, namely, semi-implicit relaxation schemes for hyperbolic systems and adaptive multiresolution algorithms, in order to achieve fast and accurate simulations of realistic two-phase flows problems in oil transportation. By ‘realistic’ we mean problems that are modelled by partial differential equation (PDE) systems closed by sophisticated thermodynamics and hydrodynamics laws, set out over a terrain-induced geometry and associated with time-dependent boundary conditions. Although the combination of these techniques is not a straightforward matter, it is made possible via a careful examination of the objectives of the simulation problem and suitable adaptations of which we shall give the details. Significant benchmarks demonstrate the efficiency of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

This work aims at numerically approximating the entropy weak solutions of Euler-like systems asymptotically recovered from the multi-pressure Navier–Stokes equations in the regime of an infinite Reynolds number. The nonconservation form of the limit PDE model makes the shock solutions to be sensitive with respect to the underlying small scales. Here we propose to model these small scale effects via a set of generalized jump conditions expressed in terms of the independent internal energies. The interest in considering internal energies stems from the presence of solely first-order nonconservative products by contrast to other variables. These nonconservative products are defined in the now classical sense proposed by Dal Maso, LeFloch and Murat. We show how to enforce the generalized jump conditions at the discrete level with a fairly simple numerical procedure. This method is proved to satisfy a full set of stability estimates and to produce approximate solutions in good agreement with exact Riemann solutions.