System Of Nonlinear Hyperbolic Equations Research Articles

Global solutions and blow-up for a coupled system of nonlinear hyperbolic equations with weak damping

Global solutions and blow-up for a coupled system of nonlinear hyperbolic equations with weak damping

On the theoretical foundation of overset grid methods for hyperbolic problems II: Entropy bounded formulations for nonlinear conservation laws

We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes a two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. In addition to the interface coupling, which is required, entropy dissipation and coupling can optionally be added through the use of linear penalties within the overlap region.

On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems Ii: Entropy Bounded Formulations for Nonlinear Conservation Laws

We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. Entropy dissipation and additional coupling in the overlap region are added through the use of linear penalties.

Necessary Conditions for the Existence of a Saddle Point in one Optimal Control Problem for Systems of Hyperbolic Equations

In this paper, we consider one optimal control problem with a multipoint quality functional described by a system of nonlinear hyperbolic equations with Goursat boundary conditions. Using a modified version of the increment method, various necessary first-order optimality conditions such as the Pontryagin maximum principle and the linearized maximum condition are proved.

Correction to: Coupled System of Nonlinear Hyperbolic Equations with Variable-Exponents: Global Existence and Stability

Correction to: Coupled System of Nonlinear Hyperbolic Equations with Variable-Exponents: Global Existence and Stability

Stability analysis of a hyperbolic stochastic Galerkin formulation for the Aw-Rascle-Zhang model with relaxation.

We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coefficients in the truncated series. Stochastic Galerkin formulations are presented in conservative form and for smooth solutions also in the corresponding non-conservative form. This allows to obtain stabilization results, when the system is relaxed to a first-order model. Computational tests illustrate the theoretical results.

Coupled System of Nonlinear Hyperbolic Equations with Variable-Exponents: Global Existence and Stability

In this paper, we consider a coupled system of two nonlinear hyperbolic equations with variable-exponents in the damping and source terms. Under suitable assumptions on the intial data and the variable exponents, we prove a global existence theorem, using the Stable-set method. Then, we establish a decay estimate of the solution energy, by Komornik’s integral approach.

On the special properties of the shear shock wave in an incompressible elastic medium without preliminary deformations

The nonlinear system of hyperbolic equations describing a helical one-dimensional motion of a nonlinear elastic incompressible medium is investigated. Types and velocities of possible divergent shock waves are determined on the basis of a joint analysis of the dynamic compatibility conditions, the characteristics of this system and the Riemann invariants along them. For the single transverse shock wave in previously undeformed medium it is shown that the shearing direction is invariant at the leading shock front determined by the initial action and independent of post-impact loading. The solution of boundary-value problem about shock loading on a circular cylindrical cavity boundary is constructed by the matched asymptotic expansions method and the ray method modified for shock waves.

Bounder solution on a strip to a system of nonlinear hyperbolic equations with mixed derivatives

Bounder solution on a strip to a system of nonlinear hyperbolic equations with mixed derivatives

Mechanisms of Formation of Natural Hydrogen Reservoirs in Thermal Aquifers - Impact of Bacteria and Gas Bubble Dynamics

Underground hydrogen reservoirs, whose existence has been confirmed recently by geologists all over the world, represent a new source of renewable energy. Hydrogen is formed as the product of reaction of serpentinization between ferromagnesian minerals of the Mantle and water in the offshore zones of subduction or in continental zones at considerable depth beneath hydrothermal aquifers. Initially dissolved in water, hydrogen can create free gas bubbles when the local concentration of H2 dissolved in water exceeds the equilibrium value. The bubbles raise up and create the gas cap of free hydrogen. This process is significantly influenced by methanogen bacteria inhibiting in aquifers and consuming hydrogen for their metabolism. The reactions initiated by bacteria leads to the appearance of other volatile components as methane. The dimension, the form of the gas cap and the concentration of hydrogen and methane determine the efficiency of gas production and the energy potential of the reservoir. In the present paper we develop the conceptual mathematical model of gas cap formation in hydrogen reservoir. The bacterial activity is described by the equation of population dynamics with specific kinetic functions obtained by the authors by treating experimental data. The formation of free gas is modelled in terms of the formation of oversaturated nuclei, their growth and simultaneous motion of isolated bubbles. As far as the bubbles grow, they stick together and create continuous free gas. Thus, the hydrogen motion consists of two stages: (i) the motion of isolated bubbles, controlled by the equations of ganglion dynamics; and (ii) the motion of continuous gas through water, governed by Darcy’s law. The bubble growth is ensured by the mass exchange with hydrogen dissolved in water, which involves a non-equilibrium in the average hydrogen concentrations. Such a problem of hydrogen raising is solved analytically in simplified situation without bacteria. The problem can be reduced to a system of nonlinear hyperbolic equations, which has either continuous or discontinuous solutions (shock waves) depending on the degree of the non-equilibrium. Multiples deformations of the shocks under gas raising are described. The impact of bacteria is studied numerically by using the open source Basilisk (developed by D’Alembert Institute). We have revealed non trivial scenarios of the appearance of piece-wise constant traveling waves, or auto-oscillations caused by the nonlinear population dynamics. The result obtained enables us to give estimation to the characteristic time of gas cap formation and the evolution of its composition in time.

Existence and asymptotic behavior for systems of nonlinear hyperbolic equations

The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.

Numerical solution of initial-boundary system of nonlinear hyperbolic equations

In this article, we present a numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral Jacobi-Gauss-Radau collocation (J-GR-C) method. A J-GR-C method in combination with the implicit Runge-Kutta scheme are employed to obtain a highly accurate approximation to the mentioned problem. J-GR-C method, based on Jacobi polynomials and Gauss-Radau quadrature integration, reduces solving the system of nonlinear hyperbolic equations to solve a system of nonlinear ordinary differential equations (SNODEs). In the examples given, numerical results by the J-GR-C method are compared with the exact solutions. In fact, by selecting relatively few J-GR-C points, we are able to get very accurate approximations. In this way, the results show that this method has a good accuracy and efficiency for solving coupled partial differential equations.

A Chebyshev-Gauss-Radau Scheme For Nonlinear Hyperbolic System Of First Order

A numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral collocation method is presented in this article. A Chebyshev-Gauss-Radau collocation (C-GR-C) method in combination with the implicit Runge- Kutta scheme are employed to obtain highly accurate approximations to the mentioned problem. The collocation points are the Chebyshev interpolation nodes. This approach reduces this problem to solve system of nonlinear ordinary differential equations which are far easier to be solved. Indeed, by selecting a limited number of colloc ation nodes, we obtain an accurate results. The numerical examples demonstrate the accuracy, efficiency, and versatility of the me thod.

Global existence and asymptotic behaviour for systems of nonlinear hyperbolic equations

We consider the initial-boundary value problem for a class of nonlinear hyperbolic equations system in a bounded domain. Using the potential well theory, the existence of global solutions is investigated. We also established the asymptotic behaviour of global solutions as t → + ∞ by applying the multiplier method.

Numerical modelling of three-phase immiscible flow in heterogeneous porous media with gravitational effects

This paper presents a new numerical formulation for the simulation of immiscible and incompressible three-phase water–gas–oil flows in heterogeneous porous media. We take into account the gravitational effects, both variable permeability and porosity of porous medium, and explicit spatially varying capillary pressure, in the diffusive fluxes, and explicit spatially varying flux functions, in the hyperbolic operator. The new formulation is a sequential time marching fractional-step procedure based in a splitting technique to decouple the equations with mixed discretization techniques for each of the subproblems: convection, diffusion, and pressure–velocity. The system of nonlinear hyperbolic equations that models the convective transport of the fluid phases is approximated by a modified central scheme to take into account the explicit spatially discontinuous flux functions and the effects of spatially variable porosity. This scheme is coupled with a locally conservative mixed finite element formulation for solving parabolic and elliptic problems, associated respectively with the diffusive transport of fluid phases and the pressure–velocity problem. The time discretization of the parabolic problem is performed by means of an implicit backward Euler procedure. The hybrid-mixed formulation reported here is designed to handle discontinuous capillary pressures. The new method is used to numerically investigate the question of existence, and structurally stable, of three-phase flow solutions for immiscible displacements in heterogeneous porous media with gravitational effects. Our findings appear to be consistent with theoretical and experimental results available in the literature.

Time evolution for a model of epidermis growth

In this paper, we study a system of nonlinear hyperbolic equations, with nonlocal boundary conditions and a free boundary, arising in the modeling of epidermis growth. The model was introduced in a previous paper (Gandolfi et al. in J Math Biol 62(1):111–141, 2010) where conditions for the existence of a steady state were investigated. The present paper is devoted to prove existence and uniqueness of a solution to the evolution problem and of the related moving boundary representing the external surface of the epidermis. The proof of the theorem is based on the integration along characteristic curves in order to obtain suitable estimates allowing to set up a fixed point procedure. The modellistic aim of the paper is a description of the structure of the epidermis as a layered aggregate of different type of cells.

High-order discontinuous element-based schemes for the inviscid shallow water equations: Spectral multidomain penalty and discontinuous Galerkin methods

Two commonly used types of high-order-accuracy element-based schemes, collocation-based spectral multidomain penalty methods (SMPM) and nodal discontinuous Galerkin methods (DGM), are compared in the framework of the inviscid shallow water equations. Differences and similarities in formulation are identified, with the primary difference being the dissipative term in the Rusanov form of the numerical flux for the DGM that provides additional numerical stability; however, it should be emphasized that to arrive at this equivalence between SMPM and DGM requires making specific choices in the construction of both methods; these choices are addressed. In general, both methods offer a multitude of choices in the penalty terms used to introduce boundary conditions and stabilize the numerical solution. The resulting specialized class of SMPM and DGM are then applied to a suite of six commonly considered geophysical flow test cases, three linear and three non-linear; we also include results for a classical continuous Galerkin (i.e., spectral element) method for comparison. Both the analysis and numerical experiments show that the SMPM and DGM are essentially identical; both methods can be shown to be equivalent for very special choices of quadrature rules and Riemann solvers in the DGM along with special choices in the type of penalty term in the SMPM. Although we only focus our studies on the inviscid shallow water equations the results presented should be applicable to other systems of nonlinear hyperbolic equations (such as the compressible Euler equations) and extendable to the compressible and incompressible Navier–Stokes equations, where viscous terms are included.

Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

We develop a new hierarchical reconstruction (HR) method [17,28] for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.

Darboux-integrable two-component nonlinear hyperbolic systems of equations

An approach based on a study of the structure of characteristic rings is used to solve the classification problem of Darboux-integrable nonlinear hyperbolic systems of equations. We classify two-component systems of equations with two integrals of the first order and two integrals of the second order. For the resulting systems of equations, x − and y − integrals are constructed and their general solutions are found.

Adomian Decomposition Method for Solving Goursat's Problems

In this paper, Goursat’s problems for: linear and nonlinear hyperbolic equations of second-order, systems of nonlinear hyperbolic equations and fourth-order linear hyperbolic equations in which the attached conditions are given on the characteristics curves are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and efficient.