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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

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.

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.

An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving the unsteady compressible Navier-Stokes equations. The time discretization is based on an explicit formulation for the convective fluxes and an implicit formulation for the viscous fluxes. The implicit procedure uses a fast iterative algorithm based on a partial uncoupling of variables in neighboring elements introduced in previous works by the authors. In the present work, the method is associated to a Newton-Krylov Jacobian-free method. This association allows to reduce memory requirements and operation counts by avoiding the complete construction of the Jacobian matrix and solving a problem of reduced size. Numerical examples in two space dimensions indicate that the performance of the present method is seen to be significantly improved in terms of CPU time.

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.

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.

We construct a `well-balanced' and `asymptotic preserving' scheme for the approximation of the model problem of gas dynamics equations with gravity and friction. The friction terms we consider are quite general. We interpret our simple Riemann solver in such a way that the expected properties are directly inherited from the properties of the system of PDEs which is approximated.

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.

In the context of systems of nonlinear conservation laws it can be crucial to use adaptive grids in order to correctly simulate the singularities of the solution over long time ranges while keeping the computing time within acceptable bounds. The adaptive space grid must vary in time according to the local smoothness of the solution. More sophisticated and recent methods also adapt the time-step locally to the space discretization according to the stability condition. We present here such a method designed for an explicit-implicit Lagrange-projection scheme, addressing physical problems where slow kinematic waves coexist with fast acoustic ones. Numerical simulations are presented to validate the algorithms in terms of robustness and efficiency.

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.