System Of Hyperbolic Differential Equations Research Articles

Thermodynamically Compatible Hyperbolic Model for Two-Phase Compressible Fluid Flow with Surface Tension

A two-phase flow model for compressible immiscible fluids is presented, the derivation of which is based on the use of the theory of symmetric hyperbolic thermodynamically compatible systems. The model is an extension of the previously proposed thermodynamically compatible model of compressible two-phase flows due to the inclusion of new state variables of the medium associated with surface tension forces. The governing equations of the model form a hyperbolic system of differential equations of the first order and satisfy the laws of thermodynamics (energy conservation and entropy increase). The properties of the model equations are studied and it is shown that the Young–Laplace law of capillary pressure is fulfilled in the asymptotic approximation at the continuum level.

Geometry of principal stress trajectories for piece‐wise linear yield criteria under axial symmetry

AbstractRigid‐ and elastic‐ plastic solids obeying quite a general yield criterion represented by linear functions of the principal stresses are considered. A general axisymmetric state of stress satisfying the hypothesis of Haar and von Karman is analyzed in quasi‐static flow. The circumferential velocity vanishes. A superimposed restriction on the yield criterion is that the system of stress equations is hyperbolic. The primary objective of this study is to select optimal coordinates and unknowns for deriving the integrable differential relations in their most convenient form for numerical treatments. It is shown that there exists a simple relation between the scale factors of the principal lines coordinate system (the coordinate curves of this coordinate system coincide with the trajectories of the principal stresses). Another simple relation exists between the scale factors and the principal stresses. The mapping between the principal lines and cylindrical coordinates is determined by a system of hyperbolic differential equations.

Interaction of a Singular Surface with a Characteristic Shock in a Relaxing Gas with Dust Particles

Abstract A system of hyperbolic differential equations outlining one-dimensional planar, cylindrical symmetric and spherical symmetric flow of a relaxing gas with dust particles is considered. Singular surface theory used to study different aspects of wave propagation and its culmination to the steepened form. The evolutionary behavior of the characteristic shock is studied. A particular solution of the governing system of equations is used to discuss the steepened wave form, characteristic shock and their interaction. The results of the interaction between the steepened wave front and the characteristic shock using the general theory of wave interaction are discussed. Also, the influence of relaxation and dust parameters on the steepened wave front, the formation of a characteristic shock, reflected and transmitted waves after interaction and a jump in shock acceleration are investigated.

Curved Geometries from Planar Director Fields: Solving the Two-Dimensional Inverse Problem.

Thin nematic elastomers, composite hydrogels, and plant tissues are among many systems that display uniform anisotropic deformation upon external actuation. In these materials, the spatial orientation variation of a local director field induces intricate global shape changes. Despite extensive recent efforts, to date there is no general solution to the inverse design problem: How to design a director field that deforms exactly into a desired surface geometry upon actuation, or whether such a field exists. In this work, we phrase this inverse problem as a hyperbolic system of differential equations. We prove that the inverse problem is locally integrable, provide an algorithm for its integration, and derive bounds on global solutions. We classify the set of director fields that deform into a given surface, thus paving the way to finding optimized fields.

A partial outer convexification approach to control transmission lines

In this paper we derive an efficient optimization approach to calculate optimal controls of electric transmission lines. These controls consist of time-dependent inflows and switches that temporarily disable single arcs or whole subgrids to reallocate the flow inside the system. The aim is then to find the best energy input in terms of boundary controls in combination with the optimal configuration of switches, where the dynamics is driven by a coupled system of hyperbolic differential equations. We use a well-known three-step optimization approach based on the idea of partial outer convexification, for which we establish that the analytical requirements for its application hold for each fixed spatial discretization of the underlying partial differential equation, provided that combinatorial constraints are only pointwise in time. A comparison with a direct solver yields very promising results, also for problems with from an application viewpoint important switch up-time and down-time constraints, which are not pointwise in time and thus not fully covered by theory.

CHARACTERISTIC PROBLEM FOR THE ONE SYSTEM OF HYPERBOLIC DIFFERENTIAL EQUATIONS OF THE THIRD ORDER

In the paper the well-posed characteristic problem is considered for the hyperbolic differential equation of the third order with nonmultiple characteristics. The regular solution of the characteristic problem for the hyperbolic differential equation of the third order with the nonmultiple characteristics is constructed in an explicit form. The well-posed characteristic problem is considered for one system of hyperbolic differential equations of the third order. The regular solution of the characteristic problem for the one system of hyperbolic differential equations of the third order is constructed. The theorem for the Hadamards well-posedness characteristic problem for the one system of hyperbolic differential equations is considered as the result of the research.

On a hyperbolic system arising in liquid crystals modeling

We consider a model of liquid crystals, based on a nonlinear hyperbolic system of differential equations, that represents an inviscid version of the model proposed by Qian and Sheng. A new concept of dissipative solution is proposed, for which a global-in-time existence theorem is shown. The dissipative solutions enjoy the following properties: (i) they exist globally in time for any finite energy initial data; (ii) dissipative solutions enjoying certain smoothness are classical solutions; (iii) a dissipative solution coincides with a strong solution originating from the same initial data as long as the latter exists.

Numerical Leak Detection in a Pipeline Network of Complex Structure with Unsteady Flow

An inverse problem for a pipeline network of complex loopback structure is solved numerically. The problem is to determine the locations and amounts of leaks from unsteady flow characteristics measured at some pipeline points. The features of the problem include impulse functions involved in a system of hyperbolic differential equations, the absence of classical initial conditions, and boundary conditions specified as nonseparated relations between the states at the endpoints of adjacent pipeline segments. The problem is reduced to a parametric optimal control problem without initial conditions, but with nonseparated boundary conditions. The latter problem is solved by applying first-order optimization methods. Results of numerical experiments are presented.

Space mapping techniques for the optimal inflow control of transmission lines

In this paper we present an efficient optimization approach for the dynamic inflow control of electric transmission lines. For the description of transmission lines several mathematical models are available. Here, we focus on the telegraph network equations which consist of a coupled system of hyperbolic differential equations. The optimization is then to find the best possible energy input in terms of boundary control to satisfy given demands. To solve this problem we introduce space mapping techniques based on a first discretize – then optimize approach. The idea of space mapping is the consideration of two model hierarchies (fine and coarse scale), where the coarse scale model is less accurate but easy to solve. We explain how this method works in the framework of inflow control, design the corresponding optimization algorithm and present various computational results to show the numerical efficiency.

Existence of Countably Many Cycles in Hyperbolic Systems of Differential Equations with Transformed Argument

We prove the existence of countably many periodic solutions of a hyperbolic system of first-order differential equations with periodic condition. The problems of existence and stability of traveling waves are investigated for a quasilinear Korteweg–de Vries equation with transformed argument and for the equation of spin combustion.

Software package for solving hyperbolic-type equations

The TURBO1 software package for solving the hyperbolic systems of differential equations on multiprocessor computing systems with distributed memory is described. The package is a standard-oriented and extensible software package. Within a single formalism, it is possible to solve several problems describing different physical processes. For their solution the package provides the user with numerical procedures and software blocks containing initial and boundary conditions, as well as mass forces specific to the problem. The potential of this package is demonstrated on an example from numerical simulation of classical problems of gas dynamics, i.e., the Rayleigh-Taylor instability (RTI) and the shear layer problems.

The Riemann problem for a hyperbolic model of incompressible fluids

The aim of the present paper is to investigate shock and rarefaction waves in a hyperbolic model of incompressible fluids. To this aim, we use the so-called extended-quasi-thermal-incompressible (EQTI) model, recently proposed by Gouin and Ruggeri (H. Gouin, T. Ruggeri, International Journal of Non-Linear Mechanics 47 (2012) 688–693). In particular, we use as constitutive equation a variant of the well-known Boussinesq approximation in which the specific volume depends not only on the temperature but also on the pressure, leading to a hyperbolic system of differential equations. The limit case of ideal incompressibility, namely when the thermal expansion coefficient and the compressibility factor vanish, is also considered.The results show that the propagation of shock waves in an EQTI fluid is characterized by small jump in specific volume and temperature, even when the jump in pressure is relevant, and rarefaction waves originating from a general Riemann problem are characterized by a very steep profile. The knowledge of the loci of the states that can be connected to a given state by a shock wave or a rarefaction wave allows also to completely solve the Riemann problem. The obtained results are confirmed by means of numerical calculations.

Eliminating Discharge Imbalances in Predicting Shallow Water Flows with Shocks

The shallow water equations are widely used in modelling environmental flows. Being a hyperbolic system of differential equations, they admit shocks that represent hydraulic jumps and bores. Although the water surface can be solved satisfactorily with the modern shock-capturing schemes, the predicted flow rate often suffers from imbalances where shocks occur, eg the mass conservation is violated by failing to maintain a constant discharge rate at every cross-section in a steady open channel flow. A total-variation-diminishing Lax-Wendroff scheme is developed, and used to demonstrate how to achieve an exact flux balance. The performance of the proposed methods is inspected through some test cases, which include 1- and 2-dimensional, flat and irregular bed scenarios. The proposed methods are shown to preserve the mass exactly, and can be easily extended to other shock-capturing models.

A property of the defining equations for the Lie algebra in the group classification problem for wave equations

We solve the group classification problem for nonlinear hyperbolic systems of differential equations. The admissible continuous group of transformations has the Lie algebra of dimension less than 5. This main statement follows from the principal property of the defining equations of the admissible Lie algebra: the commutator of two solutions is a solution. Using equivalence transformations we classify nonlinear systems in accordance with the well-known Lie algebra structures of dimension 3 and 4.

Global Cauchy problems for hyperbolic systems with characteristics admitting superlinear growth for [formula omitted

We investigate the global well-posedness of the Cauchy problem for first order linear hyperbolic systems allowing superlinear growth of the characteristic roots for | x | → + ∞ . We introduce hypotheses on the superlinear growth inspired by theorems of A. Wintner in 1945 for global solutions of ODEs and show global well-posedness of the Cauchy problem in two types of new weighted Sobolev spaces. We construct a change of the space variables of a global Liouville type which reduces to the case of bounded coefficients. As an outcome, we derive finite propagation speed and the existence of finite domains of dependence for hyperbolic systems of differential equations. We exhibit also instant blow-up of solutions near t = 0 and provide explicit examples proving that our estimates are sharp. To cite this article: D. Gourdin, T. Gramchev, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The Tychonoff-Schauder theorem and the existence of bounded solutions of quasilinear hyperbolic systems of differential equations

The Tychonoff-Schauder fixed point theorem is applied to prove the existence of bounded solutions of systems of ordinary differential equations that are C0-close to linear hyperbolic systems. Bibliography: 3 titles.

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.

NUMERICAL SOLUTION FOR HYPERBOLIC CONSERVATIVE TWO-PHASE FLOW EQUATIONS

We outline an approximate solution for the numerical simulation of two-phase fluid flows with a relative velocity between the two phases. A unified two-phase flow model is proposed for the description of the gas–liquid processes which leads to a system of hyperbolic differential equations in a conservative form. A numerical algorithm based on a splitting approach for the numerical solution of the model is proposed. The associated Riemann problem is solved numerically using Godunov methods of centered-type. Results show the importance of the Riemann problem and of centered schemes in the solution of the two-phase flow problems. In particular, it is demonstrated that the Slope Limiter Centered (SLIC) scheme gives a low numerical dissipation at the contact discontinuities, which makes it suitable for simulations of practical two-phase flow processes.

A PROBLEM FOR THE HYPERBOLIC SYSTEM OF DIFFERENTIAL EQUATIONS WITHOUT INITIAL CONDITIONS

A PROBLEM FOR THE HYPERBOLIC SYSTEM OF DIFFERENTIAL EQUATIONS WITHOUT INITIAL CONDITIONS

The curvelet representation of wave propagators is optimally sparse

AbstractThis paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [7, 9] in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width ≈ length2 at fine scales. We prove that for a wide class of linear hyperbolic differential equations, the curvelet representation of the solution operator is both optimally sparse and well organized. It is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e., faster than any negative polynomial) and well organized in the sense that the very few nonnegligible entries occur near a few shifted diagonals. Indeed, we show that the wave group maps each curvelet onto a sum of curveletlike waveforms whose locations and orientations are obtained by following the different Hamiltonian flows—hence the diagonal shifts in the curvelet representation. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles. © 2005 Wiley Periodicals, Inc.