Application of a fourth-order relaxation scheme to hyperbolic systems of conservation laws

A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourthorder central weighted essentially nonoscillatory （CWENO） reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments.

SBV REGULARITY OF GENUINELY NONLINEAR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS IN ONE SPACE DIMENSION

The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper.

EXACT CONTROLLABILITY FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS WITH VERTICAL CHARACTERISTICS

We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.

SOME RESULTS ON HYPERBOLIC SYSTEMS WITH RELAXATION

In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P（p ≥ 2） decay rate.

THE MIXED PROBLEM FOR A CLASS OF NONLINEAR SYMMETRIC HYPERBOLIC SYSTEMS WITH DISCONTINUOUS DATA

This paper studies the nonlinear mixed problem for a class of symmetric hyperbolic systems with the boundary condition satisfying the dissipative condition about discontinuous data in higher dimension spaces, establishes the local existence theorem by using the method of a prior estimates, and obtains the structure of singularities of the solutions of such problems.

Boundary Control Problems for 2 ×2 Cooperative Hyperbolic Systems with Infinite Order Operators

In this study, boundary control problems with Neumann conditions for 2 × 2 cooperative hyperbolic systems involving infinite order operators are considered. The existence and uniqueness of the states of these systems are proved, and the formulation of the control problem for different observation functions is discussed.

An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrodinger Operator

In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.

THE INTERACTIONS OF SHOCK WAVES OF NONSTRICTLY HYPERBOLIC SYSTEMS

The interaction of shock waves is investigated for the following nonstrictly hyperbolic system: [GRAPHICS] The interaction of shock waves is complicated, with new types of shock waves, and new singula rities in the dependence of interaction on the relative positions of the three states separated by shock waves. Several ideas are introduced to helo organize and clarify the new phenomena.

PIECEWISE SMOOTH SOLUTION OF THE FIRST ORDER SEMILINEAR HYPERBOLIC SYSTEMS IN HIGHER SPACE DIMENSION

It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. In 1985, J, Rauch & M. Reed have provad the existence and uniqueness of piecewise smooth solution for

MULTIDIMENSIONAL RELAXATION APPROXIMATIONS FOR HYPERBOLIC SYSTEMS OF CONSERVATION LAWS

We construct and implement a non-oscillatory relaxation scheme for multidimensional hyperbolic systems of conservation laws. The method transforms the nonlinear hyperbolic system to a semilinear model with a relaxation source term and linear characteristics which can be solved numerically without using either Riemann solver or linear iterations. To discretize the relaxation system we consider a high-resolution reconstruction in space and a TVD Runge-Kutta time integration. Detailed formulation of the scheme is given for problems in three space dimensions and numerical experiments are implemented in both scalar and system cases to show the effectiveness of the method.

Asymptotic Behavior of Global Classical Solutions to Quasilinear Hyperbolic Systems of Diagonal Form

This paper deals with the asymptotic behavior of global classical solutions to quasilinear hyperbolic systems of diagonal form with weakly linearly degenerate characteristic fields. On the basis of global existence and uniqueness of C^1 solution, we prove that the solution to the Cauchy problem approaches a combination of C^1 traveling wave solutions when t tends to the infinity.

δ-WAVE FOR 1-D AND 2-D HYPERBOLIC SYSTEMS

Here a new kind of nonlinear wave, which is called δ-wave, is described by some high resolution difference solutions for Riemann problems of one-dimensional (1-D) and two-dimensional (2-D) nonlinear hyperbolic systems in conservation laws. Some phenomena are numerically shown for the solutions of Riemann problems for 2-D gas dynamics systems

Regional gradient observability for semilinear hyperbolic systems： HUM approach

The paper aims to extend the notion of regional observability of the gradient to the semilinear hyperbolic case, in order to reconstruct the gradient of the initial conditions in a subregion w of the domain evolution Ω. We start with an asymptotically linear system, the approach is based on an extension of the Hilbert uniqueness method （HUM） and Schauder＇s fixed point theorem. The analysis leads to an algorithm which is successfully numerically implemented and illustrated with examples and simulations.

THE OPTIMAL CONVERGENCE ORDER OF THE DISCONTINUOUS FINITE ELEMENT METHODS FOR FIRST ORDER HYPERBOLIC SYSTEMS

In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems （steady and nonsteady state） is constructed and analyzed by using linear triangle elements, and the O（h^2）-order optimal error estimates are derived under the assumption of strongly regular triangulation and the Ha-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the interpolation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.

LOCAL EXISTENCE THEOREM FOR FIRST ORDER SEMILINEAR HYPERBOLIC SYSTEMS IN SEVERAL SPACE DIMENSIONS

This paper studies first order semilinear hyperbolic systems in n(n≥2)space dimensions.Under the hypothesis that the system satisfies so called`null condition',the local well posedness for its Cauchy problem with initial data in H n-12 is proved.

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws:Application to Shallow Water Equations

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form.It applies in multidimensional structured and unstructured meshes.The proposed method is an extension of the UFORCEmethod developed by Stecca,Siviglia and Toro[25],in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan.The proposed first-order method is shown to be identical to the Godunov upwindmethod in applications to a 2×2 linear hyperbolic system.The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations.Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes.Finally,numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

A Space-Time Conservative Method for Hyperbolic Systems with Stiff and Non Stiff Source Terms

In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems.We call the scheme a slope propagation(SP)method.It is an extension of our scheme derived for homogeneous hyperbolic systems[1].In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value.These small values make the relaxation term stronger and highly stiff.In such situations underresolved numerical schemes may produce spurious numerical results.However,our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved.The scheme treats the space and time in a unified manner.The flow variables and their slopes are the basic unknowns in the scheme.The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance.We use two approaches for the slope calculations of the flow variables,the first one results directly from the flux balance over the control volumes,while in the second one we use a finite difference approach.The main features of the scheme are its simplicity,its Jacobian-free and Riemann solver-free recipe,as well as its efficiency and high of order accuracy.In particular we show that the scheme has a discrete analog of the continuous asymptotic limit.We have implemented our scheme for various test models available in the literature such as the Broadwell model,the extended thermodynamics equations,the shallow water equations,traffic flow and the Euler equations with heat transfer.The numerical results validate the accuracy,versatility and robustness of the present scheme.

Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept

We present an efficient and robustmethod for stresswave propagation problems(second order hyperbolic systems)having discontinuities directly in their second order form.Due to the numerical dispersion around discontinuities and lack of the inherent dissipation in hyperbolic systems,proper simulation of such problems are challenging.The proposed idea is to denoise spurious oscillations by a post-processing stage from solutions obtained from higher-order grid-based methods(e.g.,high-order collocation or finite-difference schemes).The denoising is done so that the solutions remain higher-order(here,second order)around discontinuities and are still free from spurious oscillations.For this purpose,improved Tikhonov regularization approach is advised.This means to let data themselves select proper denoised solutions(since there is no pre-assumptions about regularized results).The improved approach can directly be done on uniform or non-uniform sampled data in a way that the regularized results maintenance continuous derivatives up to some desired order.It is shown how to improve the smoothing method so that it remains conservative and has local estimating feature.To confirm effectiveness of the proposed approach,finally,some one and two dimensional examples will be provided.It will be shown how both the numerical(artificial)dispersion and dissipation can be controlled around discontinuous solutions and stochastic-like results.

Sectorial Approach of the Gradient Observability of the Hyperbolic Semilinear Systems Intern and Boundary Cases

The aim of this paper is to study the notion of the gradient observability on a subregion w of the evolution domain W and also we consider the case where the subregion of interest is a boundary part of the system evolution domain for the class of semilinear hyperbolic systems. We show, under some hypotheses, that the flux reconstruction is guaranteed by means of the sectorial approach combined with fixed point techniques. This leads to several interesting results which are performed through numerical examples and simulations.

Distributed Control for <i>n</i>×<i>n</i>Cooperative Systems Governed by Hyperbolic Operator of Infinite Order

In this study, a distributed optimal control problem for n × n cooperative hyperbolic systems with infinite order operators and Dirichlet conditions are considered. The existence and uniqueness of the state of these systems are proved. The necessary and sufficient conditions for optimality of distributed control with constraints are found, and the set of equations and inequalities that defining the optimal control of these systems is also obtained. Finally, some examples for the control problem without constraints are given.