Hyperbolic Systems Of Balance Laws Research Articles

Sampled-data distributed control for homo-directional linear hyperbolic system with spatially sampled state measurements

This paper addresses the stability analysis problem for a class of linear hyperbolic systems with distributed controllers sampled in space and time. First, the considered system is recast in an equivalent form with a continuous time control loop and operators representing the discretization errors (spatio-temporal sampling errors). Then with the help of the Lyapunov–Razumikhin approach, the Rɛ-stability of the proposed linear hyperbolic systems of balance laws is verified via sufficient conditions. At last, the proposed method is illustrated numerically.

Shock Structure and Relaxation in the Multi-Component Mixture of Euler Fluids

The shock structure problem is studied for a multi-component mixture of Euler fluids described by the hyperbolic system of balance laws. The model is developed in the framework of extended thermodynamics. Thanks to the equivalence with the kinetic theory approach, phenomenological coefficients are computed from the linearized weak form of the collision operator. Shock structure is analyzed for a three-component mixture of polyatomic gases, and for various combinations of parameters of the model (Mach number, equilibrium concentrations and molecular mass ratios). The analysis revealed that three-component mixtures possess distinguishing features different from the binary ones, and that certain behavior may be attributed to polyatomic structure of the constituents. The multi-temperature model is compared with a single-temperature one, and the difference between the mean temperatures of the mixture are computed. Mechanical and thermal relaxation times are computed along the shock profiles, and revealed that the thermal ones are smaller in the case discussed in this study.

High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws

We introduce a general framework for the construction of well-balanced finite volume methods for hyperbolic balance laws. We use the phrase well-balancing in a broader sense, since our proposed method can be applied to exactly follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions and not only time independent solutions. The solution has to be known a priori, either as an analytical expression or as discrete data. The proposed framework modifies the standard finite volume approach such that the well-balancing property is obtained and in case the method is high order accurate, this is maintained under our modification. We present numerical tests for the compressible Euler equations with and without gravity source term and with different equations of state, and for the equations of compressible ideal magnetohydrodynamics.

Input-to-State Stability of Non-uniform Linear Hyperbolic Systems of Balance Laws via Boundary Feedback Control

In this paper, a linear hyperbolic system of balance laws with boundary disturbances in one dimension is considered. An explicit candidate Input-to-State Stability (ISS)-Lyapunov function in $$L^2-$$ norm is considered and discretised to investigate conditions for ISS of the discrete system as well. Finally, experimental results on test examples including the Saint-Venant equations with boundary disturbances are presented. The numerical results demonstrate the expected theoretical decay of the Lyapunov function.

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.

Exact Solutions of Generalized Riemann Problem for Nonhomogeneous Shallow Water Equations

In this paper, we consider quasilinear hyperbolic system of balance laws describing one-dimensional nonhomogeneous shallow water equations with generalized Riemann initial data. We obtain exact solutions to the shallow water equations with friction by using differential constraint method. A special case of the obtained solution provides well known rarefaction wave to the homogeneous case of the governing equations. We construct a convenient example for the generalized Riemann problem and study the behavior of the solution profiles.

The relative entropy method for inhomogeneous systems of balance laws

General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs. strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different characteristics are presented and the hypotheses are discussed in the setting of each example.

Two-dimensional numerical modelling of shallow water flows over multilayer movable beds

The two-dimensional modelling of shallow water flows over multi-sediment erodible beds is presented. A novel approach is developed for the treatment of multiple sediment types in morphodynamics. The governing equations include the two-dimensional shallow water equations for hydrodynamics, an Exner-type equation for morphodynamics, a two-dimensional transport equation for the suspended sediments, and a set of empirical equations for entrainment and deposition. Multilayer sedimentary beds are formed of different erodible soils with sediment properties and new exchange conditions between the bed layers are developed for the model. The coupled equations yield a hyperbolic system of balance laws with source terms. As a numerical solver for the system, we implement a fast finite volume characteristics method. The numerical fluxes are reconstructed using the method of characteristics which employs projection techniques. The proposed finite volume solver is simple to implement, satisfies the conservation property and can be used for two-dimensional sediment transport problems in non-homogeneous isotropic beds without need of complicated three-dimensional equations. To assess the performance of the proposed models, we present numerical results for a wide variety of shallow water flows over sedimentary layers. Comparisons to experimental data for dam-break problems over movable beds are also included in this study.

Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions

In this paper, we develop an accurate technique via the use of the Adomian decomposition method (ADM) to solve analytically a 2 × 2 systems of partial differential equation that represent balance laws of hyperbolic-elliptic type. We prove that the sequence of iteration obtained by ADM converges strongly to the exact solution by establishing a construction of fixed points. For comparison purposes, we also use the Sinc function methodology to establish a new procedure to solve numerically the same system. It is shown that approximation by Sinc function converges to the exact solution exponentially, also handles changes in type. A numerical example is presented to demonstrate the theoretical results. It is noted that the two methods show the symmetry in the approximate solution. The results obtained by both methods reveal that they are reliable and convenient for solving balance laws where the initial conditions are of the Riemann type.

Vanishing viscosity limit for a system of balance laws with general type initial data arising from 1D Saint-Venant model

We study the vanishing viscosity limit of a non-strictly hyperbolic system of balance laws known as the 1D Saint-Venant model whose solution admits δ-waves. The balance term depends on both the variables x and t and is unbounded in the space variable x unlike the work of Olenik [Usp Math. Nauk. 12(3), 3–73 (1957) (in Russian)]. We also studied the large time behavior when the viscosity parameter is positive. When the balance term depends only on the time variable, we obtained an explicit formula for the solution.

Wave scattering in hyperbolic balance laws and the zero relaxation limit

In the context of a simple hyperbolic system of balance laws that relaxes to a scalar conservation law, the paper investigates the process by which the synergy of waves propagating with speeds akin to the characteristic speeds of the parent system produces, in the zero relaxation limit, waves propagating with speed akin to the characteristic speed of the equilibrium conservation law.

Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

We introduce a Darcy-scale model to describe compressible multi-component flow in a fully saturated porous medium. In order to capture cross-diffusive effects between the different species correctly, we make use of the Maxwell--Stefan theory in a thermodynamically consistent way. For inviscid flow the model turns out to be a nonlinear system of hyperbolic balance laws. We show that the dissipative structure of the Maxwell-Stefan operator permits to guarantee the existence of global classical solutions for initial data close to equilibria. Furthermore it is proven by relative entropy techniques that solutions of the Darcy-scale model tend in a certain long-time regime to solutions of a parabolic limit system.

Local Proportional-Integral Boundary Feedback Stabilization for Quasilinear Hyperbolic Systems of Balance Laws

The proportional-integral (PI) boundary stabilization of nonlinear hyperbolic systems of balance laws is investigated for the $H^2$-norm, in which the control and output measurements are all located at the boundaries. The boundary conditions of the system are subject to unknown constant disturbances. The induced closed-loop system is proven to be locally exponentially stable with respect to the steady states. To this end, a set of matrix inequalities is given by constructing a new Lyapunov function as a weighted $H^2$-norm of the classical Cauchy solution and the integral of boundary output. Furthermore, the traffic flow dynamics of a freeway section are modeled with the Aw--Rascle--Zhang model. To stabilize the oscillations of traffic demand, a local PI boundary feedback controller is designed with integration of on-ramp metering and variable speed limit control. The exponential convergence of the nonlinear traffic flow dynamics in an $H^2$ sense is achieved and validated with simulations.

Stability Analysis for A Class of Linear Hyperbolic System of Balance Laws with Sampled-data Control

Abstract The stability for a class of linear hyperbolic systems with distributed sampled-data controllers is discussed in this paper. The original sampled-data system is firstly transformed into a new equivalent system by modelling the sampling induced error as a reset integrator operator. Then we construct an appropriate Lyapunov function and obtain sufficient conditions for the Re - stability of the system based on linear matrix inequalities (LMIs). A numerical example illustrates our results: when the sampling interval is within the allowable range, the solution of the system converges from the domain of attraction to a positive invariant set.

A positivity-preserving central-upwind scheme for isentropic two-phase flows through deviated pipes

Directional drilling in oil and gas extraction can encounter difficulties such as accumulation of solids in deviated pipes. Motivated by such phenomenon, we consider a model for isentropic two-phase flows through deviated pipes. The system of partial differential equations is aimed at simulating the dynamics between a particle bed and a gas phase. The pipe can be either horizontal or vertically deviated where the effects of gravity are incorporated. Furthermore, the acceleration or deceleration due to friction between phases is investigated and spectral properties of the hyperbolic system of balance laws are described. The existence and characterization of steady states under appropriate conditions is analyzed. A new type of steady states arises when a balance between gas and solid phases results in a non-uniform solid particle bed and vanishing solid velocity. This state corresponds to an accumulation of sedimented solids. A central-upwind scheme that preserves the positivity of the gas and solid densities and volume fractions is presented. Including an application of the model to an analysis of accumulation of solids, a variety of numerical tests is presented to show the merits of the scheme.

Convergence rate from hyperbolic systems of balance laws to parabolic systems

It is proved recently that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish error estimates between the smooth solutions of the hyperbolic systems of balance laws and those of the parabolic limit systems in one space dimension. The proof of the error estimates uses a stream function technique together with energy estimates. As applications of the results, we give five examples arising from physical models.

On the structure of BV entropy solutions for hyperbolic systems of balance laws with general flux function

The paper describes the qualitative structure of BV entropy solutions of a general strictly hyperbolic system of balance laws with characteristic field either piecewise genuinely nonlinear or linearly degenerate. In particular, we provide an accurate description of the local and global wave-front structure of a BV solution generated by a fractional step scheme combined with a wave-front tracking algorithm. This extends the corresponding results in [S. Bianchini and L. Yu, Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension, Comm. Partial Differential Equations 39(2) (2014) 244–273] for strictly hyperbolic system of conservation laws.

Plane-strain waves in nonlinear elastic solids with softening

Propagation of elastic waves in damaged media (concrete, rocks) is studied theoretically and numerically. Such materials exhibit a nonlinear behavior, with long-time softening and recovery processes (slow dynamics). A constitutive model combining Murnaghan hyperelasticity with the slow dynamics is considered, where the softening is represented by the evolution of a scalar variable. The equations of motion in the Lagrangian framework are detailed. These equations are rewritten as a nonlinear hyperbolic system of balance laws, which is solved numerically using a finite-volume method with flux limiters. Numerical examples illustrate specific features of nonlinear elastic waves, as well as the effect of the material’s softening. In particular, the generation of solitary waves in a periodic layered medium is illustrated numerically.

Uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries

We present a Stochastic Finite Volume - ADER (SFV-ADER) methodology for Uncertainty Quantification (UQ) in the general framework of systems of hyperbolic balance laws with uncertainty in parameters. The resulting method is weakly intrusive, meaning that deterministic solvers need minor modifications to include random contributions, and has no theoretical accuracy barrier. An illustration of the second-order version for the viscous Burgers equation with uncertain viscosity coefficient is first given in detail, under different deterministic initial conditions; the attainment of the theoretically expected convergence rate is demonstrated empirically; results and features of our scheme are discussed. We then extend the SFV-ADER method to a non-linear hyperbolic system with source terms that models one-dimensional blood flow in arteries, assuming uncertainties in a set of parameters of the problem. Results are then compared with measurements in a well-defined 1:1 experimental replica of the network of largest arteries in the human systemic circulation. For a 99%-confidence level, severe influence of parameter fluctuations is seen in pressure profiles, while secondary effects on flow rate profiles become visible, particularly in terminal branches. Results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws.

High-gain observers for a class of 2 × 2 quasilinear hyperbolic systems with 2 different velocities

In this work we extend recently proposed observer designs based on high-gain to a more general first order quasilinear hyperbolic system of balance laws. This class of systems is written in an observable form with two states, two different characteristic velocities and distributed measurement. The exponential stability of the related observation error is fully established by means of Lyapunov-based analysis.