Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

GLOBALLY SMOOTH SOLUTIONS TO CAUCHY PROBLEM OF A QUASILINEAR HYPERBOLIC SYSTEM ARISING IN BIOLOGY

This article considers Cauchy problem u(t) - (uv)(x) = 0, v(t) - u(x) = 0, u(x, 0) = u(0) (x) > 0, v(x, 0) = v(0)(x). A necessary and sufficient condition in guaranteeing that Cauchy problem admits a global C-1-solution on t greater than or equal to 0 is obtained.

GLOBAL EXISTENCE OF WEAKLY DISCONTINUOUS SOLUTIONS TO A KIND OF MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.

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.

CONVERGENCE OF THE VISCOSITY METHOD FOR A NONSTRICTLY HYPERBOLIC SYSTEM

A convergence theorem for the method of artificial viscosity applied to the nonstrictly hyperbolic system u(t)+1/2(3u2+v2)x=0, v(t)+(uv)x=0 is established. Convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves.

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.

A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT AXISYMMETRIC VESSELS

We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisyminetric vessels. Early models derived are nonconservative and/or nonho- mogeneous with measure source terms, which are endowed with infinitely many Riemann solutions for some Riemann data. In this paper, we derive a one-dimensional hyperbolic system that is conservative and homogeneous. Moreover, there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data, under a natural stability entropy criterion. The Riemann solutions may consist of four waves for some cases. The system can also be written as a 3 × 3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.

EXISTENCE OF GLOBAL L~∞ SOLUTIONS TO A GENERALIZED n × n HYPERBOLIC SYSTEM OF LEROUX TYPE

In this article, we give the existence of global L∞bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of Le Roux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2×2 to n×n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1= 0} is another difficulty.We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.

Global Structure Stability of Riemann Solution of Quasilinear Hyperbolic System of Conservation Law in Presence of Boundary:Shocks and Contact Discontinuities

In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n × n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the pieeewise C^1 solution u = u（t, x） of the problem, which can be regarded as a perturbation of the corresponding Riemann problem, is globally similar to that of the solution u = U（x/t） of the corresponding Riemann problem. The piecewise C^1 solution u = u（t, x） to this kind of problems is globally structure-stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.

Absorbing Boundary Conditions for Hyperbolic Systems

This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions.We prove the strict well-posedness of the resulting initial boundary value problem in 1D.Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme.Hereby,we have to extend the classical proofs,since the(discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.

THE CAUCHY PROBLEMS FOR HIGH DIMENSIONAL QUASILINEAR HYPERBOLIC SYSTEMS

In this paper, we prove the existence of the global smooth solution to the Cauchy problems for a class of diagonalizable high dimensional quasilinear hyperbolic systems consisted of n-equations.

A Linear Hyperbolic System with Stiff Source Terms

We consider the behavior of solutions to the system ut ＋vx =0, vt ＋ux =（au-v）/τ as the relaxation parameter τ approaches 0. If the sub characteristic condition｜a｜ is satisfied, then u approaches a solution to the equilibrium equation ut ＋ aux = 0 and v approaches au. If this condition is violated then solutions have not Fourier transformation when τ→0.

Global Classical Solutions of Inhomogeneous Quasilinear Hyperbolic Systems

In this paper,we consider a kind of quasilinear hyperbolic systems with inhomogeneous terms satisfying dissipative condition or matching condition.For the Cauchy problem of this kind of systems,we prove that,if the initial data is small and satisfies some decay condition,and the system is weakly linearly degenerate,then the Cauchy problem admits a unique global classical solution on t ≥ 0.

Analysis of Singularity for Reducible Quasi-linear Hyperbolic Systems

In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.

Blow Up of Periodic Solutions to a Class of Quasilinear Hyperbolic Systems in Diagonal Form

In this paper, we discuss the blow-up of periodic solutions to a class of quasilinear hyperbolic systems in diagonal form, and make the accurate estimate of life-span. These results in this paper extend the conclusion [1-3].