Non-conservative Hyperbolic System Research Articles

A unified consistent source term computational algorithm for the [formula omitted]-based compressible multi-fluid flow model

Using the criterion of one-fluid preservation, we developed a consistent algorithm for the source term to solve the γ-based compressible multi-fluid flow model with three approximate Riemann solvers, namely the Lax–Friedrichs (LxF), Kurganov, and Harten–Lax–van Leer contact (HLLC) solvers. The consistent algorithm comprises a standard Godunov solver with a high-order reconstruction and a consistent source term integration part. We prove that the present algorithm is consistent with Abgrall's criterion of the moving-material-interface property in the finite volume method framework. The cell boundary velocity for the source term discretization is found to be the same as HLLC solvers from five-equation model, but for the first time for LxF and Kurganov Riemann solvers. We develop 12 compressible multi-fluid solvers by combining four reconstruction schemes and three approximate Riemann solvers together with their consistent-source-term integration algorithms. All 12 solvers can maintain the one-fluid preservation and moving-material-interface properties numerically in several one- and two-dimensional example cases. The simulation of an underwater explosion demonstrates that a boundary-variation-diminishing reconstruction predicts an interface with width controlled within approximately three cells which is independent of the Riemann solver. The simulation of an explosion near a free surface demonstrates that the proposed model can simulate a severe compressible multi-fluid flow involving an interface across which there are large differences in density, pressure and parameters in the equation of state. In conclusion, the proposed consistent algorithm provides a unified framework for one kind of non-conservative hyperbolic system into a conservative hyperbolic system and a source term with velocity divergence, where the former can be computed by classical Godunov-type algorithms and the latter can be solved by the proposed consistent algorithm.

High order numerical methods for flows with hysteretic fluxes

A second order method and a limited variation WENO numerical scheme are proposed for the Scanning Hysteresis Model (SHM) which describes continuum flows with hysteretic looping fluxes with local memory. The SHM is a nonconservative hyperbolic system with discontinuous coefficients. The limited variation WENO method is fifth order accurate in spatial coordinate and third order accurate in time steps in smooth and monotone regions. The proposed high resolution methods produce the desired solutions more efficiently than the first order upwinding method. These methods are total variation diminishing for the flux, and monotonicity-preserving for the flux and the hysteresis state variable. The flux and the hysteresis state variable produced by these two methods satisfy maximum principle. These two methods also have invariant regions.

A bedload transport in shallow water model applied for sediment transport in open channel flows

This paper presents the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modelled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a non-conservative hyperbolic system. To discretize it, first we consider a Non-Homogeneous Riemann Solver scheme as well as a variant based on the use of flux limiters. In order to develop second-order scheme, we use a MUSCL method incorporating slope limiters in the spatial approximation and a two-step Runge-Kutta method for time integration. The comparison between results based on the proposed scheme and analytical results shows good agreement..

Fifth-order A-WENO schemes based on the path-conservative central-upwind method

We develop fifth-order A-WENO finite-difference schemes based on the path-conservative central-upwind method for nonconservative one- and two-dimensional hyperbolic systems of nonlinear PDEs. The main challenges in development of accurate and robust numerical methods for the studied systems come from the presence of nonconservative products. Semi-discrete second-order finite-volume path-conservative central-upwind (PCCU) schemes recently proposed in Castro Díaz et al. (2019) [8] provide one with a reliable Riemann-problem-solver-free numerical method for nonconservative hyperbolic system. In this paper, we extend the PCCU schemes to the fifth-order of accuracy in the framework of A-WENO finite-difference schemes.We apply the developed schemes to the two-layer shallow water equations. We ensure that the developed schemes are well-balanced in the sense that they are capable of exactly preserving “lake-at-rest” steady states. We illustrate the performance of the new fifth-order schemes on a number of one- and two-dimensional examples, where one can clearly see that the proposed fifth-order schemes clearly outperform their second-order counterparts.

Exact solution for Riemann problems of the shear shallow water model

The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks, rarefactions, shear and contact waves. The notion of weak solution is based on a path but the choice of the correct path is not known for this problem. In this paper, we construct exact solution for the Riemann problem assuming a linear path in the space of conserved variables, which is also used in approximate Riemann solvers. We compare the exact solutions with those obtained from a path conservative finite volume scheme on some representative test cases.

A new well-balanced finite-volume scheme on unstructured triangular grids for two-dimensional two-layer shallow water flows with wet-dry fronts

In this study, a novel two-dimensional finite-volume method on unstructured triangular meshes for two-layer shallow water flows is developed. First, the one-dimensional relaxation approach developed in Abgrall and Karni (2009) [1] is extended to the currently studied two-dimensional two-layer shallow water equations to build an unconditionally hyperbolic system. Next, a two-dimensional finite-volume HLL scheme on triangular grids for such a non-conservative hyperbolic system is constructed. Special attention is paid to guaranteeing the well-balanced property even in the presence of the wet-dry fronts. To this end, techniques such as spatial reconstruction for the secondary variables, special cell classifications for the two-layer fluid system, and special local reconstruction of auxiliary surface variables are proposed. These special techniques are further combined with new discrete formulas for the nonconservative products and the bed-slope terms, and it thus leads to an exactly well-balanced numerical scheme for the studied two-layer system even in the presence of wet-dry fronts. The developed numerical model is second order accurate and preserves the nonnegative layer depths. The performances of the proposed numerical model are investigated by several numerical experiments.

The resonant cases and the Riemann problem for a model of two-phase flows

The resonant cases in the Riemann problem for a model of two-phase flows with resonance are investigated, which complete the construction of Riemann solutions. The model is given by a nonconservative hyperbolic system of balance laws. The phase decomposition method is developed to construct solutions in each phase in a relatively separated way, but still glued to each other via the contact discontinuities. Composite wave curves can be built and intersections of wave curves in each phase plane help to determine various configurations of Riemann solutions. In particular, solutions containing multiple waves propagating with the same shock speed are constructed.

On solving the Riemann problem for non-conservative hyperbolic systems of partial differential equations

The paper addresses a new method for approximately solving the Riemann problem for an arbitrary non-conservative hyperbolic system of partial differential equations. The method is based on the reconstruction of the complete wave structure in the Riemann problem with using the Borel measures interpretation (path-representation) of the non-conservative products. In contrast to alternative approximate methods of the Riemann problem solution, e.g., HLL-type methods, the method proposed takes into account the complete wave structure of the solution and requires the calculation of only eigenvalues and eigenvectors of the matrix of the system to be considered and therefore can be easy realized for any hyperbolic system. In the case of conservative systems, the method gives the exact Riemann problem solution with any required accuracy. It can be applied, for example, to compute the exact Rieman problem solution for compressible Euler equations with an arbitrary equation of state. The method is applied to solving the Rieman problem for the Euler equations, the two-phase Baer-Nunziato equations, the elasto-plastic equations. The results obtained concern the convergence and accuracy propreties of the method.

Existence and Computation of Solutions of a Model of Traffic Involving Hysteresis

The meaning of weak solutions of a nonconservative hyperbolic system with discontinuous coefficients modeling traffic flows involving hysteresis is defined. An upwinding approximation scheme for the model is shown to be total variation diminishing. Vehicles' speeds and drivers' hysteresis states satisfy maximum and minimum principles. The limit of a convergent subsequence generated by the approximation scheme is shown to be a weak solution of the model if it is piecewise $C^1$, establishing the existence of such solutions.

A 1D numerical model for the simulation of unsteady and highly erosive flows in rivers

This work is focused on a numerical finite volume scheme for the coupled shallow water-Exner system in 1D applications with arbitrary geometry. The mathematical expressions modeling the hydrodynamic and morphodynamic components of the physical phenomenon are treated to deal with cross-section shape variations and empirical solid discharge estimations. The resulting coupled equations can be rewritten as a non-conservative hyperbolic system with three moving waves and one stationary wave to account for the source terms discretization. Moreover, the wave celerities for the coupled morpho-hydrodyamical system depend on the erosion-deposition mechanism selected to update the channel cross-section profile. This influence is incorporated into the system solution by means of a new parameter related to the channel bottom variation celerity. Special interest is put to show that, even for the simplest solid transport models as the Grass law, to find a linearized Jacobian matrix of the system can be a challenge in presence of arbitrary shape channels. In this paper a numerical finite volume scheme is proposed, based on an augmented Roe solver, first order accurate in time and space, dealing with solid transport flux variations caused by the channel geometry changes. Channel cross-section variations lead to the appearance of a new solid flux source term which should be discretized properly. The stability region is controlled by wave celerities together with a proper reconstruction of the approximate local Riemann problem solution, enforcing positive values for the intermediate states of the conserved variables. Comparison of the numerical results for several analytical and experimental cases demonstrates the effectiveness, exact well-balancedness and accuracy of the scheme.

Structure of the hydraulic jump in convergent radial flows

We are interested in the modelling of multi-dimensional turbulent hydraulic jumps in convergent radial flow. To describe the formation of intensive eddies (rollers) at the front of the hydraulic jump, a new model of shear shallow water flows is used. The governing equations form a non-conservative hyperbolic system with dissipative source terms. The structure of equations is reminiscent of generic Reynolds-averaged Euler equations for barotropic compressible turbulent flows. Two types of dissipative term are studied. The first one corresponds to a Chézy-like dissipation rate, and the second one to a standard energy dissipation rate commonly used in compressible turbulence. Both of them guarantee the positive definiteness of the Reynolds stress tensor. The equations are rewritten in polar coordinates and numerically solved by using an original splitting procedure. Numerical results for both types of dissipation are presented and qualitatively compared with the experimental works. The results show both experimentally observed phenomena (cusp formation at the front of the hydraulic jump) as well as new flow patterns (the shape of the hydraulic jump becomes a rotating square).

Coupled method for the numerical simulation of 1D shallow water and Exner transport equations in channels with variable cross-section

This work is focused on the a numerical finite volume scheme for the resulting coupled shallow water-Exner system in 1D applications with arbitrary geometry. The mathematical expression modeling the the hydrodynamic and morphodynamic components of the physical phenomenon are treated to deal with cross-section shape variations and empirical solid discharge estimations. The resulting coupled system of equations can be rewritten as a nonconservative hyperbolic system with three moving waves and one stationary wave to account for the source terms discretization. But, even for the simplest solid transport models as the Grass law, to find a linearized Jacobian matrix of the system can be a challenge if one considers arbitrary shape channels. Moreover, the bottom channel slope variations depends on the erosion-deposition mechanism considered to update the channel cross-section profile. In this paper a numerical finite volume scheme is proposed, based on an augmented Roe solver (first order accurate in time and space) and dealing with solid transport flux variations caused by the channel geometry changes. Channel crosssection variations lead to the appearance of a new solid flux source term which should be discretized properly. Comparison of the numerical results for several analytical and experimental cases demonstrate the effectiveness, exact wellbalanceness and accuracy of the scheme.

Self-organized hydrodynamics with congestion and path formation in crowds

A continuum model for self-organized dynamics is numerically investigated. The model describes systems of particles subject to alignment interaction and short-range repulsion. It consists of a non-conservative hyperbolic system for the density and velocity orientation. Short-range repulsion is included through a singular pressure which becomes infinite at the jamming density. The singular limit of infinite pressure stiffness leads to phase transitions from compressible to incompressible dynamics. The paper proposes an Asymptotic-Preserving scheme which takes care of the singular pressure while preventing the breakdown of the Courant–Friedrichs–Lewy (CFL) stability condition near congestion. It relies on a relaxation approximation of the system and an elliptic formulation of the pressure equation. Numerical simulations of impinging clusters show the efficiency of the scheme to treat congestions. A two-fluid variant of the model provides a model of path formation in crowds.

Numerical Simulations of a Nonconservative Hyperbolic System with Geometric Constraints Describing Swarming Behavior

The Vicsek model is a very popular individual based model which describes collective behavior among animal societies. A large-scale limit of the Vicsek model has been derived in [Math. Models Methods Appl. Sci., 18 (2008), pp. 1193–1215], leading to a macroscopic version of the model. In this work, we want to numerically validate this macroscopic Vicsek (MV) model. However, there is no standard theory to study analytically or numerically the MV model since it is a nonconservative hyperbolic system with a geometric constraint. Different formulations of the MV model are presented and lead to several nonequivalent numerical schemes. In particular, we derive a numerical scheme, denoted by the splitting method, based on a relaxation of the geometric constraint. To test the veracity of these schemes, we compare the simulations of the macroscopic and microscopic models with each other. The numerical simulations reveal that the microscopic and macroscopic models are in good agreement, provided that we use the spl...

Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes

In this paper, we study the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modeled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a nonconservative hyperbolic system. To discretize it, first we consider a Roe-type first order scheme as well as a variant based on the use of flux limiters. These first order schemes are then extended to second order accuracy by means of a new MUSCL-type reconstruction operator on unstructured meshes. Finally, some numerical tests are presented.

A numerical study of a particular non-conservative hyperbolic problem

We study the ability of several numerical schemes to solve a non-conservative hyperbolic system arising from a flow simulation of solid–liquid–gas slurries with the so-called virtual mass effect. Two classes of numerical schemes are used: some Roe-type finite volume schemes, which are based on the resolution of linearized Riemann problems, and some (centered or upwind) schemes with an additional artificial diffusion, such as the classical Rusanov scheme. For flow regimes of interest (steady as well as unsteady flows), the computational process breaks down for some schemes. Indeed, for such flows, the system has at least one eigenvalue having a small magnitude in the interior of the computational domain and this is a possible reason for the failure of some upwind schemes using the resolution of a linearized Riemann problem. Such a failure does not appear with, for instance, the Rusanov scheme which is well known for its robustness. Furthermore, since the system is non-conservative, it is not clear what a weak solution is, when the solution is discontinuous (at least, one needs to have the non-conservative equivalent of the Rankine–Hugoniot jump conditions) and we show that the approximate solution given by different numerical schemes converges towards different “weak solutions”.

Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods

This paper is concerned with the numerical approximation of bedload sediment transport due to water evolution. For the hydrodynamical component we consider Shallow Water equations. The morphodynamical component is defined by a continuity equation, which is defined in function of the solid transport discharge. We present several deterministic models, such as Meyer-Peter & Müller, Van Rijn or Grass model. We also present an unified definition for the solid transport discharge, and we compare with Grass model. Both components define a coupled system of equations that can be rewrite as a non-conservative hyperbolic system. To discretize it, we consider finite volume methods with or without flux limiters and high order state reconstructions. Finally we present several tests, where we observe numerically the order of the numerical schemes. Comparisons with analytical solutions and experimental data are also presented.

Schéma nonlinéaire pour l'approximation numérique d'un système hyperbolique non conservatif

We consider, for numerical approximations, a nonconservative hyperbolic system which arises when modeled a bifluid flow. We introduce a diffusion tensor to define the discontinuous shock wave solutions and we exhibit additional laws satisfied by the smooth solutions. Arguing the additional laws, we propose a numerical method shown to be consistant with the definition of discontinuous solutions. To cite this article: C. Berthon, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 1069–1072.

An approximate solution of the Riemann problem for a realisable second-moment turbulent closure

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.

An Exact Riemann Solver for Multicomponent Turbulent Flow

The main objective of this paper is to provide some adequate way to compute the non-conservative hyperbolic system which describes a multicomponent turbulent flow. The model is written for an isentropic gas. The exact solution of the Riemann Problem (RP) associated to the hyperbolic system is exhibited. It is composed of constant states separated by rarefaction waves, or shock waves and a contact discontinuity. The selection of the admissible part of the shock curve is obtained using an entropy criterion. This entropy is the total energy of the system. Thanks to the latter, one may compute the exact solution of the Riemann problem, assuming genuinely non linear fields contain sufficiently weak shocks.