Hyperbolic Systems Of Balance Laws Research Articles

Global BV entropy solutions to the Gauss–Codazzi system

The Gauss–Codazzi system for the given metric whose Gauss curvature is negative, is considered in the present paper. We provide some suitable conditions on the given metric such that the system admits a global BV entropy solution. Our main approach is applying the method of seeking global BV entropy solutions to the hyperbolic system of balance laws to the Gauss–Codazzi system. The global existence of BV solutions to the Gauss–Codazzi system yields a C1,1 isometric immersion of two dimensional manifold into R3.

Junction-Generalized Riemann Problem for stiff hyperbolic balance laws in networks: An implicit solver and ADER schemes

In this paper we design a new implicit solver for the Junction-Generalized Riemann Problem (J-GRP), which is based on a recently proposed implicit method for solving the Generalized Riemann Problem (GRP) for systems of hyperbolic balance laws. We use the new J-GRP solver to construct an ADER scheme that is globally explicit, locally implicit and with no theoretical accuracy barrier, in both space and time. The resulting ADER scheme is able to deal with stiff source terms and can be applied to non-linear systems of hyperbolic balance laws in domains consisting on networks of one-dimensional sub-domains. In this paper we specifically apply the numerical techniques to networks of blood vessels. We report on a test problem with exact solution for a simplified network of three vessels meeting at a single junction, which is then used to carry out a systematic convergence rate study of the proposed high-order numerical methods. Schemes up to fifth order of accuracy in space and time are implemented and tested. We then show the ability of the ADER scheme to deal with stiff sources through a numerical simulation in a network of vessels. An application to a physical test problem consisting of a network of 37 compliant silicon tubes (arteries) and 21 junctions, reveals that it is imperative to use high-order methods at junctions, in order to preserve the desired high order of accuracy in the full computational domain. For example, it is demonstrated that a second-order method throughout, gives comparable results to a method that is fourth order in the interior of the domain and first order at junctions. We are concerned with stiff hyperbolic balance laws in networks.An implicit solver for the junction-generalized Riemann problem is proposed.ADER schemes of arbitrary accuracy in space and time for networks are constructed.Convergence rates studies of schemes up to fifth order are carried out.It is necessary to match the accuracy at junctions to that of the rest of the domain.

Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids

The problem of sub-shock formation within a shock structure solution of hyperbolic systems of balance laws is investigated for a binary mixture of multi-temperature Eulerian fluids. The main purpose of this work is the analysis of the ranges of Mach numbers characterizing shock-structure solutions with different features, continuous or not, and to show the existence of ranges, below the maximum unperturbed characteristic velocity, for which each constituent of the mixture may develop a sub-shock within a smooth shock structure profile. The theoretical results are supported by numerical calculations.

Hyperbolic balance laws with relaxation

This expository paper surveys the progress in a research program aiming at establishing the existence and long time behavior of $BV$ solutions to the Cauchy problem for hyperbolic systems of balance laws modeling relaxation phenomena.

Singular Perturbation Approximation of Linear Hyperbolic Systems of Balance Laws

This technical note deals with a class of linear hyperbolic systems of balance laws with multiple time scales. The scale of time constants is modeled by a perturbation parameter. This parameter is introduced in both dynamics and boundary conditions. The solution of the full system is approximated by that of the reduced subsystem when the perturbation parameter is small enough. Lyapunov technique is used to prove it. The main result is illustrated by an academic example. Moreover, the boundary control synthesis to a gas flow transport model is shown based on singular perturbation approach.

Mathematical entropy and Euler–Cattaneo–Maxwell system

In this paper, we introduce a notion of the mathematical entropy for hyperbolic systems of balance laws with (not necessarily symmetric) relaxation. As applications, we deal with the Timoshenko system, the Euler–Maxwell system and the Euler–Cattaneo–Maxwell system. Especially, for the Euler–Cattaneo–Maxwell system, we observe that its dissipative structure is of the regularity-loss type and investigate the corresponding decay property. Furthermore, we prove the global existence and asymptotic stability of solutions to the Euler–Cattaneo–Maxwell system for small initial data.

A remark on the Glimm scheme for inhomogeneous hyperbolic systems of balance laws

General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small BV data under appropriate assumptions on the decay of the flux and the source with respect to space and time. There is neither a hypothesis about equilibrium solution nor about the dependence of the source on the state vector as previous results have assumed.

Abstract State-Space Models for a Class of Linear Hyperbolic Systems of Balance Laws

The paper discusses and compares different abstract state-space representations for a class of linear hyperbolic systems defined on a one-dimensional spatial domain. It starts with their PDE representation in both weakly and strongly coupled forms. Next, the homogeneous state equation including the unbounded formal state operator is presented. Based on the semigroup approach, some results of well-posedness and internal stability are given. The boundary and observation operators are introduced, assuming a typical configuration of boundary inputs as well as pointwise observations of the state variables. Consequently, the homogeneous state equation is extended to the so-called boundary control state/signal form. Next, the classical additive statespace representation involving (A, B, C)-triple of state, input and output operators is considered. After short discussion on the appropriate Hilbert spaces, state-space equation in the so-called factor form is also presented. Finally, the resolvent of the system state operator A is discussed.

Electric transmission lines: Control and numerical discretization

We are interested in the mathematical modeling and control of electrical power lines. The electric transmission lines under consideration are modeled as directed graph. The evolution of voltage and current is described by a linear 2 2 system of hyperbolic balance laws. Open and closed loop control concepts are investigated for control actions at nodal points in the network. For the modeling of electric transmission lines several mathematical approaches can be found in the literature [1, 3]. A typical control problem is to find the optimal power flow when allowing for power supply at selected nodes. Mathematically, this leads to possibly large scale finite-dimensional nonlinear optimization problem [4]. From those, simplified models are derived, which use linear or mixed-integer programming techniques to determine the optimal flow [7]. Based on those models extensions to oscillator networks are used to capture some effects of time-dependent transmission lines. A typical model for the evolution is given by the spatially one-dimensional telegraph equation supplemented by suitable source terms. For instance, in [9] the power and signal losses in communication systems are determined by the telegraph equation. Depending on the sheeted material sometimes lossless transmission lines may be considered. In this case the telegraph equation reduces to the linear wave equation for the voltage. Mathematically, closed loop control questions for the wave equation have extensively been studied for example in [10]. Closed loop control of a single transmission line including source terms has been established using a Lyapunov function approach. We extend the existing result to more general networks and introduce coupling conditions for the telegraph equation as well. The idea is to derive a closed loop network control relying heavily on the construction of the suitable Lyapunov function for the telegraph equation. Furthermore, we introduce an open loop control concept and numerically compare the arising control actions. This extends recent investigations conducted for other applications [3, 11, 12].

Implicit, semi-analytical solution of the generalized Riemann problem for stiff hyperbolic balance laws

We present a semi-analytical, implicit solution to the generalized Riemann problem (GRP) for non-linear systems of hyperbolic balance laws with stiff source terms. The solution method is based on an implicit, time Taylor series expansion and the Cauchy–Kowalewskaya procedure, along with the solution of a sequence of classical Riemann problems. Our new GRP solver is then used to construct locally implicit ADER methods of arbitrary accuracy in space and time for solving the general initial–boundary value problem for non-linear systems of hyperbolic balance laws with stiff source terms. Analysis of the method for model problems is carried out and empirical convergence rate studies for suitable tests problems are performed, confirming the theoretically expected high order of accuracy.

Well-balanced central-upwind scheme for a fully coupled shallow water system modeling flows over erodible bed

Intense sediment transport and rapid bed evolution are frequently observed under highly-energetic flows, and bed erosion sometimes is of the same magnitude as the flow itself. Simultaneous simulation of multiple physical processes requires a fully coupled system to achieve an accurate hydraulic and morphodynamical prediction. In this paper, we develop a high-order well-balanced finite-volume method for a new fully coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution.The nonequilibrium sediment transport equation is used to predict the sediment concentration variation. Since bed-load, sediment entrainment and deposition have significant effects on the bed evolution, an Exner-based equation is adopted together with the Grass bed-load formula and sediment entrainment and deposition models to calculate the morphological process. The resulting 5×5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. A special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. The proposed fully coupled model is verified on a number of numerical experiments.

Asymptotic behavior of BV solutions to hyperbolic systems of balance laws with relaxation

For hyperbolic systems of balance laws governing relaxation processes, in one space dimension, with source incurring nonnegative entropy production and satisfying a Kawashima-type condition, it is shown that when the initial data have small total variation on (-∞, ∞) and decay rapidly to zero, as |x| → ∞, then the Cauchy problem possesses a unique admissible BV solution, in the large, with total variation decaying to zero, as t → ∞.

Finite-volume schemes for Friedrichs systems with involutions

In applications solutions of systems of hyperbolic balance laws often have to satisfy additional side conditions. We consider initial value problems for the general class of Friedrichs systems where the solutions are constrained by differential conditions given in the form of involutions. These occur in particular in electrodynamics, electro- and magnetohydrodynamics as well as in elastodynamics. Neglecting the involution on the discrete level typically leads to instabilities.To overcome this problem in electrodynamical applications it has been suggested in Munz et al. (2000) to solve an extended system. Here we suggest an extended formulation to the general class of constrained Friedrichs systems. It is proven for explicit Finite-Volume schemes that the discrete solution of the extended system converges to the weak solution of the original system for vanishing discretization and extension parameter under appropriate scalings. Moreover we show that the involution is weakly satisfied in the limit. The proofs rely on a reformulation of the extension as a relaxation-type approximation and careful use of the convergence theory for finite-volume methods for systems of Friedrichs type. Numerical experiments illustrate our analytical results.

The Optimal Decay Estimates on the Framework of Besov Spaces for Generally Dissipative Systems

We give a new decay framework for general dissipative hyperbolic system and hyperbolic-parabolic composite system, which allow us to pay less attention on the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood-Paley pointwise energy estimates and new time-weighted energy functionals to establish the optimal decay estimates on the framework of spatially critical Besov spaces for degenerately dissipative hyperbolic system of balance laws. Based on the $L^{p}(\mathbb{R}^{n})$ embedding and improved Gagliardo-Nirenberg inequality, the optimal $L^{p}(\mathbb{R}^{n})$-$L^{2}(\mathbb{R}^{n})(1\leq p<2)$ decay rates and $L^{p}(\mathbb{R}^{n})$-$L^{q}(\mathbb{R}^{n})(1\leq p<2\leq q\leq\infty)$ decay rates are further shown. Finally, as a direct application, the optimal decay rates for 3D damped compressible Euler equations are also obtained.

Periodic BV solutions of hyperbolic balance laws with dissipative source

We establish the existence and long time behavior of spatially periodic BV solutions to strictly hyperbolic systems of balance laws with partially dissipative source satisfying the Kawashima condition.

Stability and convergence of relaxation schemes to hyperbolic balance laws via a wave operator

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505–527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357–1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277] for systems of hyperbolic conservation laws without source terms.

Global Classical Solutions to the Mixed Initial-boundary Value Problem for a Class of Quasilinear Hyperbolic Systems of Balance Laws

Global Classical Solutions to the Mixed Initial-boundary Value Problem for a Class of Quasilinear Hyperbolic Systems of Balance Laws

Initial–boundary value problems for a system of hyperbolic balance laws arising from chemotaxis

This paper is devoted to the analytical study of initial–boundary value problems for a system of hyperbolic balance laws derived from a repulsive chemotaxis model with logarithmic sensitivity. In the first part of the paper we show that, subject to the Dirichlet boundary conditions, classical solutions exist globally in time for large initial data. Asymptotically in time, the solutions are shown to converge to their boundary data at an exponential rate as time goes to infinity. Numerical simulations are supplied to corroborate the analytical results. The analytic approach developed herein can be utilized to handle a family of initial and initial–boundary value problems of the model and related models with similar mathematical structure. As a demonstration of the effectiveness of our approach, in the second part of the paper we show that, subject to the Neumann–Dirichlet boundary conditions, classical solutions exist and converge to constant equilibrium states for large initial data and for arbitrary values of the chemical diffusion coefficient. This improves a previous result obtained in [30] where the smallness of the chemical diffusion coefficient was required.

Traveling waves in porous media with phase-dependent damping

We consider a simple model for the fluid flow in a porous medium. The model consists of a hyperbolic system of balance laws, which take into account phase changes and allow for metastable states thanks to the introduction of an equilibrium pressure. A damping term is included as well, which depend not only on the velocity but also on the phase of the fluid; in particular, it vanishes in the vapor phase. The existence and uniqueness of traveling waves is proved in several important cases.

BV Solutions of Hyperbolic Balance Laws with Relaxation in the Absence of Conserved Quantities

The paper establishes the existence and long time behavior of BV solutions to a hyperbolic system of balance laws with partially dissipative source, manifesting relaxation. The component of the state vector that is actively damped decays to zero, while the complementary component, whose total mass is not necessarily conserved, behaves asymptotically as a heat kernel.