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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

We are interested in the numerical approximation of the discontinuous solutions of non-conservative hyperbolic systems. We more precisely consider a non-conservative formulation of the usual gas dynamics equations and show how to slightly modify the so-called Roe-type path-conservative schemes to properly capture the underlying shock discontinuities. Numerical evidences are proposed. The present note follows a first comment on the computation of non-conservative products in [1] .

In this work we propose a high-order discretization of the Baer-Nunziato two-phase flow model (Baer and Nunziato, Int. J. Multiphase Flow, 12 (1986), pp. 861-889) with closures for interface velocity and pressure adapted to the treatment of discontinuous solutions, and stiffened gas equations of states. We use the discontinuous Galerkin spectral element method (DGSEM), based on collocation of quadrature and interpolation points (Kopriva and Gassner, J. Sci. Comput., 44 (2010), pp. 136-155). The DGSEM uses summation-by-parts (SBP) operators in the numerical quadrature for approximating the integrals over discretization elements (Carpenter et al., SIAM J. Sci. Comput., 36 (2014), pp. B835-B867; Gassner et al., J. Comput. Phys., 327 (2016), pp. 39-66). Here, we build upon the framework provided in (F. Renac, J. Comput. Phys., 382 (2019), pp. 1-36) for nonconservative hyperbolic systems to modify the integration over cell elements using the SBP operators and replace the physical fluxes with entropy conservative fluctuation fluxes from Castro et al. (SIAM J. Numer. Anal., 51 (2013), pp. 1371-1391), while we derive entropy stable fluxes applied at interfaces. This allows to establish a semi-discrete inequality for the cell-averaged physical entropy, while being high-order accurate. The design of the numerical fluxes also formally preserves the kinetic energy at the semi-discrete level. High-order integration in time is performed using strong stability-preserving Runge-Kutta schemes and we propose conditions on the numerical parameters for the positivity of the cell-averaged void fraction and partial densities. The positivity of the cell-averaged solution is extended to nodal values by the use of an a posteriori limiter. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.

International audience; We consider a class of kinetic models for which a moment equation has a natural interpretation. We show that, depending on their velocity field, some models lead to moment equations that enable one to compute monokinetic solutions economically. We detail the example of a multiscale structured cell population model, consisting of a system of 2D transport equations. The reduced model, a system of 1D transport equations, is obtained by computing the moments of the 2D model with respect to one variable. The 1D solution is defined from the solution of the 2D model starting from an initial condition that is a Dirac mass in the direction removed by reduction. Long time properties of the 1D model solution are obtained in connection with properties of the support of the 2D solution for general case initial conditions. Finite volume numerical approximations of the 1D reduced model can be used to compute the moments of the 2D solution with proper accuracy. The numerical robustness is studied in the scalar case, and a full scale vector case is presented.

This paper presents an analysis of refinement indicators for the simulation of steady and unsteady flows by means of p -adaptive algorithms for discontinuous Galerkin (DG) methods. Residual-error, discretization-error and feature-based indicators are compared by studying their effect on convergence history, computational gain and refinement regions selected by the adaptive algorithm. The analysis is initially carried out on steady flow configurations. Static p -adaptive simulations of the periodic flow past a cylinder at R e = 100 are then performed. Compared to uniform p -refinement, a reduction between 50% and 75% in the number of degrees of freedom is obtained for all test cases considered. The accuracy and efficiency observed for the small-scale energy density [1] and spectral decay [2] indicators demonstrate their great potential for the adaptive simulation of unsteady flows.

39 pages; International audience; This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well–balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem.

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.

In this work, we propose a numerical method to handle discontinuous fluxes arising in transport-like equations. More precisely, we study hyperbolic PDEs with flux transmission conditions at interfaces between subdomains where coefficients are discontinuous. A dedicated finite volume scheme with a limited high order enhancement is adapted to treat the discontinuities arising at interfaces. The validation of the method is done on one- and two-dimensional toy problems for which exact solutions are available, allowing us to do a thorough convergence study. We then apply the method to a biological model focusing on complex cell dynamics that initially motivated this study and illustrates the full potentialities of the scheme.

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 could have been given the title: "How to positively and implicitly solve Euler equations using only linear scalar advections." The new relaxation method we propose is able to solve Euler-like systems—as well as initial and boundary value problems—with real state laws at very low cost, using a hybrid explicit-implicit time integration associated with the Arbitrary Lagrangian-Eulerian formalism. Furthermore, it possesses many attractive properties, such as: (i) the preservation of positivity for densities; (ii) the guarantee of min-max principle for mass fractions; (iii) the satisfaction of entropy inequality, under an expressible bound on the CFL ratio. The main feature that will be emphasized is the design of this optimal time-step, which takes into account data not only from the inner domain but also from the boundary conditions.

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.

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.

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.

This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent entropies. Various models for complex compressible materials typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the generic impossibility of recasting equivalently the present system in full conservation form. Classical finite volume methods are shown to grossly fail in the capture of viscous shock solutions that are of primary interest in the present work. To enforce for validity a set of generalized jump conditions that we introduce, we propose a systematic and effective correction procedure, the so-called nonlinear projection method, and prove that it preserves all the stability properties satisfied by suitable Godunov-type methods. Numerical experiments assess the relevance of the method when exhibiting approximate solutions in close agreement with exact solutions.

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.

We analyse the uniqueness of the solutions of a PDE system from the framework of compressible turbulent models. Smoothness properties of the turbulent viscosity closure law are of central importance. Several closures of practical importance, including the most widely used law, indeed fail to be Lipschitz continuous in the natural neighborhood of a null turbulent energy. For such models, we prove the existence of infinitely many distinct traveling wave solutions which exhibit positive turbulent energy but connect at infinity end states with vanishing turbulence. Examples and counter-examples are given.

In several papers of Bouchut, Bourdarias, Perthame and Coquel, Le Floch ((13), (7)...), a general methodology has been developed to construct second order flnite volume schemes for hyperbolic systems of conservation laws satisfying the entire family of entropy inequalities. This approach is mainly based on the construction of an entropy diminishing projection. Unfor- tunately, the explicit computation of this projection is not always easy. In the flrst part of this paper, we carry out this computation in the important case of the Euler equations of gas dynamics. In the second part, we present several numerical applications of the projection in the context of flnite volume schemes.

An approximate solution of the Riemann problem associated with a realisable and objective turbulent second-moment closure, which is valid for compressible flows, is examined. The main features of the continuous model are first recalled. An entropy inequality is exhibited, and the structure of waves asso- ciated with the non-conservative hyperbolic convective system is briefly described. Using a linear path to connect states through shocks, approximate jump conditions are derived, and the existence and uniqueness of the one-dimensional Riemann problem solution is then proven. This result enables to construct exact or approximate Riemann-type solvers. An approximate Riemann solver, which is based on Gallouet's recent proposal is eventually presented. Some computations of shock tube problems are then discussed.

42 pages; We construct an approximate Riemann solver for the isentropic Baer-Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.

International audience; We give in this paper a short review of some recent achievements within the framework of multiphase flow modeling. We focus first on a class of compressible two-phase flow models, detailing closure laws and their main properties. Next we briefly summarize some attempts to model two-phase flows in a porous region, and also a class of compressible three-phase flow models. Some of the main difficulties arising in the numerical simulation of solutions of these complex and highly non-linear systems of PDEs are then discussed, and we eventually show some numerical results when tackling two-phase flows with mass transfer.

We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems which is based on the so-called thin interface model and uses an augmented formulation and an additional unknown for the interface location; this framework has the advantage of avoiding any explicit modeling of the interface structure. In the present paper, we pursue our investigation of the augmented formulation and we introduce a new coupling framework which is now based on the so-called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe that the Cauchy problem is well-posed. Then, our main achievement in the present paper is the design of a new well-balanced finite volume scheme which is adapted to the thick interface model, together with a proof of its convergence toward the unique entropy solution (for a broad class of nonlinear hyperbolic equations). Due to the presence of a possibly resonant interface, the standard technique based on a total variation estimate does not apply, and DiPerna's uniqueness theorem must be used. Following a method proposed by Coquel and LeFloch, our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the discrete entropy dissipation in the proposed scheme., Comment: 21 pages

We establish the convergence of the finite volume method applied to multidimensional hyperbolic conservation laws and based on monotone numerical flux-functions. Our technique applies with a fairly unrestrictive assumption on the triangulations (“flat elements” are allowed) and to Lipschitz continuous flux-functions. We treat the initial and boundary value problem and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of Kruzkov. The proof of convergence is based on a convergence framework [Coquel and LeFloch, Math. Comp., 57 (1991), pp. 169–210 and J. Numer. Anal., 30 (1993), pp. 675–700]. From a convex decomposition of the scheme, we derive a new estimate for the rate of entropy dissipation and a new formulation of the discrete entropy inequalities. These estimates are shown to be sufficient for the passage to the limit in the discrete equation. Convergence follows from DiPerna’s uniqueness result in the class of entropy measure-valued solutions.

In the present work, we propose a fractional step method for computing approximate solutions of the isentropic Baer-Nunziato two-phase flow model. The scheme relies on an operator splitting method corresponding to a separate treatment of fast propagation phenomena due to the acoustic waves on the one hand and slow propagation phenomena due to the fluid motion on the other. The scheme is proved to preserve positive values of the statistical fractions and densities. We also provide two test-cases that assess the convergence of the method. Nous proposons ici une méthode à pas fractionnaires pour le calcul de solutions approchées pour la version isentropique du modèle diphasique de Baer-Nunziato. Le schéma s’appuie sur un splitting de l’opérateur temporel correspondant à la prise en compte différenciée des phéno-mènes de propagation rapide dus aux ondes acoustiques et des phénomènes de propagation lente dus aux ondes matérielles. On prouve que le schéma permet de préserver des valeurs positives pour les taux statistiques de présence des phases ainsi que pour les densités. Deux cas tests numériques permettent d’illustrer la convergence de la méthode.

International audience; Phase transition problems in compressible media can be modelled by mixed hyperbolicelliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. In recent years several tracking-type algorithms have been suggested for numerical approximation. Typically a core piece of these algorithms is the usage of exact Riemann solvers incorporating the kinetic relation at the location of phase boundaries. However, exact Riemann solvers are computationally expensive or even not available. In this paper we present a class of approximate Riemann solvers for hyperbolic-elliptic models that relies on a generalized relaxation procedure. It preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions. In combination with a novel sub-iteration procedure the approximate Riemann solvers are used in the tracking algorithms. The efficiency of the approach is validated on a barotropic system with linear kinetic relation where exact Riemann solvers are available. For a nonlinear kinetic relation and a thermoelastic system we use the new method to gain information on the Riemann problem. Up to our knowledge an exact solution for arbitrary Riemann data is currently not available in these cases.

International audience; We introduce a class of two-fluid models that complies with a few theoretical requirements that include: (i) hyperbolicity of the convective subset, (ii) entropy inequality, (iii) uniqueness of jump conditions for non-viscous flows. These specifications are necessary in order to compute relevant approximations of unsteady flow patterns. It is shown that the Baer-Nunziato model belongs to this class of two-phase flow models, and the main properties of the model are given, before showing a few numerical experiments.

International audience; Abstract: An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving nonlinear second-order partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The procedure uses an iterative algorithm with reduced evaluation cost. The size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low-order and high-order degrees of freedom. Numerical examples are presented for the nonlinear convection-diffusion equation in one and two dimensions including steady and unsteady flow problems. The performance of the present method is investigated in terms of CPU time and compared to a fully implicit method. A Von Neumann stability analysis is carried out in order to determine the stability and damping properties of the method. Besides a fairly reduced CPU effort, numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method.

This paper presents the design and implementation of a numerical method to simulate a multiscale model describing the selection process in ovarian follicles. The PDE model consists in a quasi-linear hyperbolic system of large size, namely Nf × Nf, ruling the time evolution of the cell density functions of Nf follicles (in practice Nf is of the order of a few to twenty). These equations are weakly coupled through the sum of the first order moments of the density functions. The time-dependent equations make use of two structuring variables, age and maturity, which play the roles of space variables. The problem is naturally set over a compact domain of R2. The formulation of the time-dependent controlled transport coefficients accounts for available biological knowledge on follicular cell kinetics. We introduce a dedicated numerical scheme that is amenable to parallelization, by taking advantage of the weak coupling. Numerical illustrations assess th e relevance of the proposed method both in term of accuracy and HPC achievements. Ce document présente la conception et l’implémentation d’une méthode numérique servant à simuler un modèle multiéchelle décrivant le processus de sélection des follicules ovariens. Le modèle EDP consiste en un système hyperbolique quasi linéaire de grande taille, typiquement Nf × Nf, gouvernant l’évolution des fonctions de densité cellulaire pour Nf follicules (en pratique Nf est de l’ordre de quelques-uns à une vingtaine). Ces équations d’évolution utilisent deux variables structurantes, l’âge et la maturité, qui jouent le rôle de variables d’espace. Le problème est naturellement posé sur un domaine compact de R2. La formulation du transport à coefficients variables au cours du temps en fonction du contrôle est issue des connaissances disponibles sur la cinétique cellulaire au sein des follicules ovariens. Nous présentons un schéma numérique dédié au problème parallélisable en tirant parti du faible couplage. Des simulations numériques permettent d’évaluer la pertinence de la méthode proposée tant en terme de précision que de calcul haute performance.

We present a new approximate Riemann solver (ARS) for the gas dynamics equations in Lagrangian coordinates and with general nonlinear pressure laws. The design of this new ARS relies on a generalized Suliciu pressure relaxation approach. It gives by construction the exact solutions for isolated entropic shocks and we prove that it is positive, Lipschitz-continuous and satisfies an entropy inequality. Finally, the ARS is used to develop either a classical entropy conservative Godunov-type method, or a Glimm-type (random sampling-based Godunov-type) method able to generate infinitely sharp discrete shock profiles. Numerical experiments are proposed to prove the validity of these approaches.

In this paper, an L ∞ ( L 1 ) {L^\infty }({L^1}) -error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order h 1 / 4 {h^{1/4}} , where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for which no estimate of the total variation and of the modulus of continuity in time are needed. The result is new even for the finite volume method constructed from monotone numerical flux functions.

International audience; We consider the seven-equation model for compressible two-phase flows and propose a large time-step numerical scheme based on a time implicit-explicit Lagrange-Projection strategy introduced by Coquel et al. for Euler equations. The main objective is to get a Courant-Friedrichs-Lewy (CFL) condition driven by (slow) contact waves instead of (fast) acoustic waves.

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce here an augmented formulation which allows for the modeling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial value problem which need to be supplemented with further admissibility conditions. This first paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch, and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions which apply to resonant wave patterns., Comment: 28 pages

A general framework is proposed for proving convergence of high-order accurate difference schemes for the approximation of conservation laws with several space variables. The standard approach deduces compactness from a BV (bounded variation) stability estimate and Helly's theorem. In this paper, it is proved that an a priori estimate weaker than a BV estimate is sufficient. The method of proof is based on the result of uniqueness given by Di Perna in the class of measure-valued solutions. Several general theorems of convergence are given in the spirit of the Lax-Wendroff theorem. This general method is then applied to the high-order schemes constructed with the modified-flux approach.

International audience; This work is devoted to the coupling of two fluid models, such as two Euler systems in Lagrangian coordinates, at a fixed interface. We define coupling conditions which can be expressed in terms of continuity of some well chosen variables and then solve the coupled Riemann problem. In the present setting where the interface is charactersitic, a particular choice of dependent variables which are transmitted ensures the overall conservativity.

International audience; We present a numerical scheme based on a two-step convection-relaxation strategy for the simulation of compressible two-phase flows with phase change. The core system used here is a simple isothermal model where stiff source terms account for mass transfer.

International audience; Closure laws for interfacial pressure and interfacial velocity are proposed within the frame work of two-pressure two-phase flow models. These enable us to ensure positivity of void fractions, mass fractions and internal energies when investigating field by field waves in the Riemann problem.; On propose des lois de fermeture de vitesse et de pression d’interface pour un modèle d’écoulement diphasique à deux pressions. Celles-ci assurent champ par champ de respecter la positivité des fractions volumiques, des variables densité et énergie interne si on examine le problème de Riemann.

This special issue comprises selected papers from the workshop New Trends in Model Coupling, Theory, Numerics and Applications (NTMC'09) which took place in Paris, September 2 - 4, 2009. The research of optimal technological solutions in a large amount of industrial systems requires to perform numerical simulations of complex phenomena which are often characterized by the coupling of models related to various space and/or time scales. Thus, the so-called multi-scale modelling has been a thriving scientific activity which connects applied mathematics and other disciplines such as physics, chemistry, biology or even social sciences. To illustrate the variety of fields concerned by the natural occurrence of model coupling we may quote: meteorology where it is required to take into account several turbulence scales or the interaction between oceans and atmosphere, but also regional models in a global description, solid mechanics where a thorough understanding of complex phenomena such as propagation of cracks needs to couple various models from the atomistic level to the macroscopic level; plasma physics for fusion energy for instance where dense plasmas and collisionless plasma coexist; multiphase fluid dynamics when several types of flow corresponding to several types of models are present simultaneously in complex circuits; social behaviour analysis with interaction between individual actions and collective behaviour.

In this paper, we propose a new way to perform second-order enhancement using slope-limiters for the numerical linear advection ∂tΨ + u∂xΨ =0 , while ensuring not only the monotonicity of each component of Ψ, but also that of a nontrivial combination G(Ψ) called control variable. This problem arises in fluid mechanics codes using the Lagrange-Projection formalism, such as [1], where the additional monotonicity property is required by physical considerations within the “remap” step. The sophisticated slope reconstruction procedure we devised can be extended to a large number of industrial problems.

We are concerned with the numerical simulation of two-phase flows representing oil transportation along a 1-D pipeline. More specifically, we wish to show that, for this kind of problems, it is highly recommended to make use of the Lagrange-Projection formalism. Indeed, this approach naturally splits the effects of fast (acoustic) and slow (kinematic) waves, thus enabling us to design numerical schemes with many desirable properties: explicit with respect to slow waves and implicit with respect to fast waves. The design of the overall explicit-implicit Lagrange-Projection method involves many interesting features, among which the positivity of the partial densities is of utmost interest to us. In this paper, the emphasis will be laid on the motivation and the general philosophy of the Lagrange-Projection method we propose. Extensive numerical results for realistic test cases are then presented in order to illustrate the capabilities of the method