Value Problem For Hyperbolic Systems Research Articles

Computational algorithms based on the fundamental solution of task operator

An approximation of matrix Green’s function for a hyperbolic system of linear differential equations of the first order has been created. An algorithm for an approximate solution of the generalized Riemann problem of discontinuity decay in the presence of additional conditions at the boundaries has been proposed. A computational algorithm for the approximate solution of boundary value problems for hyperbolic systems of equations has been set up on the basis of these solutions. The algorithm is implemented for a system of equations describing the propagation of elastic waves in a block-fractured medium. The results of the numerical experiments aimed at studying the possible mechanisms of propagation of elastic waves to a considerable depth without scattering have been presented.

WKB EXPANSIONS FOR HYPERBOLIC BOUNDARY VALUE PROBLEMS IN A STRIP: SELFINTERACTION MEETS STRONG WELL-POSEDNESS

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip $\mathbb{R}^{d-1}\times [0,1]$. In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the sides of the strip. This phenomenon is not new and has already been studied in Benoit (Geometric optics expansions for hyperbolic corner problems, I: self-interaction phenomenon, Anal. PDE9(6) (2016), 1359–1418), Sarason and Smoller (Geometrical optics and the corner problem, Arch. Rat. Mech. Anal.56 (1974/75), 34–69) in the corner geometry. In this framework the existence of such selfinteracting rays is linked to specific geometries of the characteristic variety and may seem to be somewhat anecdotal. However for the strip geometry such rays become generic. The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness (Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip (preprint)). It is interesting to point here that such a situation already occurs in the half space geometry (Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math.23 (1970), 277–298).

Existence of a Solution to a System of Partial Differential Algebraic Equations of Arbitrary Index

We consider a linear nonstationary system of first order partial differential equations that is not resolved with respect to the derivatives and identically degenerates in the domain. Without using the change of variables, we construct the structural form whose set of solutions coincides with the set of solutions to the original system. We obtain the hyperbolicity conditions and conditions for the correctness of initial and boundary conditions. We establish the existence of solutions to the initial-boundary value problem for hyperbolic systems of differential algebraic equations.

On a Priori Energy Estimates for Characteristic Boundary Value Problems

Motivated by the study of certain non linear free-boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the solution to certain boundary value problems, in the conormal Sobolev space $$H^1_{ tan}$$ , can be transformed into an $$L^2$$ a priori estimate of the same problem.

Asymptotic of solutions for second IBVP for hyperbolic systems in non-smooth domains

The purpose of this paper is to study the asymptotic behaviour of solutions of the second initial boundary value problem for hyperbolic systems in an infinite cylinder with the base containing conical points. Some new results on the asymptotics of solutions of elliptic problem depending on a parameter are also given.

THE FIRST INITIAL BOUNDARY VALUE PROBLEM FOR HYPERBOLIC SYSTEMS IN INFINITE NONSMOOTH CYLINDERS

This paper is concerned with the first initial boundary value problem for hyperbolic systems in infinite cylinders with the nonsmooth base. Some results on the regularity of generalized solutions of this problem are given.

On the regularity of the solution for the second initial boundary value problem for hyperbolic systems in domains with conical points

Abstract In this paper, we deal with the second initial boundary value problem for higher order hyperbolic systems in domains with conical points. We establish several results on the well-posedness and the regularity of solutions.

THE FIRST INITIAL BOUNDARY VALUE PROBLEM FOR HYPERBOLIC SYSTEMS IN INFINITE NONSMOOTH CYLINDERS

This paper is concerned with the first initial boundary value problem for hyperbolic systems in infinite cylinders with the nonsmooth base. Some results on the regularity of generalized solutions of this problem are given.

Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem, J. Differential Equations 222 (2006) 515–549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697–715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.

Decay Estimates for Hyperbolic Balance Laws

This work is concerned with time-decay properties of small-amplitude global smooth solutions to the initial value problem for hyperbolic systems of balance laws admitting an entropy and satisfying the stability condition. By using energy methods in both the physical space and the Fourier space, we obtain the optimal decay estimates of solutions and their derivatives in the L^2 -norm up to order s-1 , provided that the initial data are in H^s . A key ingredient in our analysis is a time-weighted energy estimate, leading to a decay estimate for multi-dimensional problems without assuming the L^1 property on initial data.

Existence of Solutions for Hyperbolic System of Second Order Outside a Domain

We study the mixed initial-boundary value problem for hyperbolic system of second order outside a closed domain. The existence of solutions to this problem is proved and the estimate for the regularity of solutions is given. The application of the existence theorem to elastrodynamics is discussed.

On the asymptotics of solutions of the first initial boundary value problem for hyperbolic systems in infinite cylinders with base containing conical points

The purpose of this paper is to study the asymptotic behavior of solutions of the first initial boundary value problem for hyperbolic systems in an infinite cylinder with the base containing conical points. Some results on the asymptotics of the solutions of an elliptic problem depending on a parameter are given.

Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws II

Kurganov and Tadmor have developed a numerical scheme for solving the initial value problem for hyperbolic systems of conservation laws. They showed that in the scalar case their scheme satisfies a local maximum–minimum principle i.e., the solution at future is bounded above and below by the solution at current locally. In this paper we show that this scheme is positive in the sense of Friedrichs for systems as well. We present the scheme of Kurganov and Tadmor as a convex combination of composites of positive schemes. Since each component of a composite scheme is bounded in the l 2 norm, so is the convex combination of the composites. To achieve second order accuracy in time, we use a Runge–Kutta type scheme due to Shu and Osher. We present two numerical experiments to add to the ones carried out by Kurganov and Tadmor.

The Green Matrix for Strictly Hyperbolic Systems with Second-Order Derivatives

Mathematical modeling of dynamic problems of continuum mechanics in domains of complicated geometry for various types of boundary conditions usually leads to boundary value problems for hyperbolic systems. The method of boundary integral equations is a convenient tool for solving such problems; it allows one to reduce the original differential boundary value problem in a domain to a system of boundary integral equations on the boundary, thus reducing the dimension of the equations, improving the stability of numerical solution methods, and so on. Now this method is widely used for a large class of problems in mechanics and mathematical physics. The construction of fundamental solutions and kernels of boundary integral equations of the system is a key point of the method of boundary integral equations. The fundamental solutions of classical equations of elliptic and hyperbolic types have been studied quite comprehensively [1, 2]. Less is known about fundamental solutions of systems of partial differential equations. Related papers mainly deal with specific equations of continuum mechanics [3–7]. In the present paper, we consider strictly hyperbolic systems of M equations with second-order derivatives in an (N + 1)-dimensional space. For such systems, we construct the Green matrix and analyze its properties. We also study systems invariant under the orthogonal group.

Hyperbolic reductions for Einstein's equations

We consider the problem of reducing initial value problems for Einstein's field equations to initial value problems for hyperbolic systems, a problem of importance for numerical as well as analytical investigations of gravitational fields. The main steps and the most important objectives in designing hyperbolic reductions are discussed. Various reductions which have already been studied in the literature or which can easily be derived from previous discussions of the field equations are pointed out and some of their specific features are indicated. We propose new reductions based on the use of the Bianchi equation for the conformal Weyl tensor. These reductions involve symmetric hyperbolic systems of propagation equations and allow a number of different gauge conditions. They use unknowns in a most economic way, supplying direct and non-redundant information about the geometry of the time slicing and the four-dimensional spacetime. Some of this information is directly related to concepts of gravitational radiation. All these reductions can be extended to include the conformal field equations. Those which are based on the ADM representation of the metric can be rewritten in flux conserving form.

On the spectrum of evolution operators generated by hyperbolic systems

We prove that the asymptotic behavior of semigroups generated by mixed-initial value problems for hyperbolic systems in one space variable is determined by the zeroes of the characteristic equation. The proof is based on the following reduction principle: we construct a reduced system (which is much simpler than the given one) such that the difference of the semigroups is a compact operator. The construction of the reduced system is presented in the non-autonomous case because it has implifications in the study of systems which are time periodic.

On the behavior over long time intervals of finite difference and finite element approximations to first order hyperbolic systems

Standard error estimates for approximations to hyperbolic equations are only valid over a finite time interval. This paper considers the behavior of the error in the two most common discretization schemes over the entire interval 0 ⩽ t < ∞. Finite difference approximations are analyzed for approximating the Cauchy problem for hyperbolic systems. It is recalled that, if the system is dissipative, error estimates that are global in t can be obtained. For strictly conservative systems, the role of local energy decay is pointed out and used to obtain global in t local in x error estimates. Finite element methods are considered for nonlinear, initial, boundary value problems for hyperbolic systems. It is proven that if the nonlinear term is locally Lipschitz and the equation is dissipative when linearized about the true solution, then global in t convergence follows with the expected rates.

Numerical relativity. I. The characteristic initial value problem

In this, the first of a series of papers on numerical relativity, the characteristic initial value problem is posed in a form suitable for numerical integration. It can be reduced to the solution of two initial value problems for sets of ordinary differential equations (on the initial surfaces) and the solution of two initial value problems for hyperbolic systems of equations, one linear, one quasilinear. The initial data may be specified freely. Subsequent papers will develop numerical solutions of Einstein’s equations with use of this formalism.

On the Stability of Galerkin Methods for Initial-Boundary Value Problems for Hyperbolic Systems

The stability of approximating the solution of mixed initial-boundary value problems for hyperbolic systems by semidiscrete Galerkin methods is studied. It is shown that a particular straightforward Galerkin method yields an unstable approximation, and that this numerical instability is caused by an improper treatment of the boundary. Stable schemes are then presented, one of which differs from the unstable scheme only insofar as the treatment of the boundary is concerned. These stable schemes make use of a particular matrix which symmetrizes the differential system. It is therefore shown that the use of this matrix is crucial to the stability of the computations as well as for obtaining a priori bounds on the energy of the continuous system. This symmetrizing matrix is also related to the diagonalizing matrix for the system of hyperbolic equations and to the Lyapunov matrix for the system of ordinary differential equations resulting from the application of Galerkin’s method.

Extrapolation applied to certain discretization methods solving the initial value problem for hyperbolic differential equations

The procedurediwiex presented in this paper provides an approximate solution to Cauchy's initial value problem for general hyperbolic systems of first order. The procedurecharex can be applied to the initial value problem for a hyperbolic system of quasi-linear differential equations. This second method is a kind of method of characteristics. It produces a solution for the whole domain of determinancy. Both procedures use extrapolation to the limit.