Interactions of delta shock waves for a class of nonstrictly hyperbolic system of conservation laws

In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter $\varepsilon$ tend to zero, we analyze the stability of Riemann solutions.     

关于一个非线性几何光学中的非严格双曲守恒律系统的研究

研究了一个产生于非线性几何光学中的非严格双曲守恒律系统.该系统具有强非线性流函数项,且狄拉克激波可能同时出现在解的两个状态变量中.通过未知函数的一个变换,该系统的非线性流函数项得到弱化,从而其黎曼问题被完全解决.     

Solution of a system of nonstrictly hyperbolic conservation laws

In this paper we study a special case of the initial value problem for a 2×2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class ofL ^∞ functions always but may contain δ-measures as well: Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution.     

An adaptive wavelet viscosity method for hyperbolic conservation laws

We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre‐orthogonal spline‐)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least‐squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

Solutions to quasilinear hyperbolic conservation laws with initial discontinuities

We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t T~*.Then, when T~*≤ t T′, we get a new shock wave between two rarefactions, and then, when t T′,another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t →∞. The technique does not involve dimensional reduction or coordinate transformation.     

Boundary layer to a system of viscous hyperbolic conservation laws

In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for n × n hyperbolic system of conservation laws with artificial viscosity in the half line （0, ∞）. We first show that a boundary layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an elementary energy method.     

一类非严格双曲系统的Riemann解的稳定性

在Riemann初值的小扰动意义下,对于一类非严格双曲系统证明Riemann解是稳定的.通过详细分析基本波的相互作用,利用特征分析方法研究扰动的Riemann解的全局结构以及解的大时间性态.     

Solutions to a hyperbolic system of conservation laws on two boundaries

This paper studies the interaction of elementary waves including delta-shock waves on two boundaries for a hyperbolic system of conservation laws. The solutions of the initialboundary value problem for the system are constructively obtained. In the problem the initialboundary data are in piecewise constant states.     

On the optimization of flux limiter schemes for hyperbolic conservation laws

A classic strategy to obtain high‐quality discretizations of hyperbolic partial differential equations is to use flux limiter (FL) functions for blending two types of approximations: a monotone first‐order scheme that deals with discontinuous solution features and a higher order method for approximating smooth solution parts. In this article, we study a new approach to FL methods. Relying on a classification of input data with respect to smoothness, we associate specific basis functions with the individual smoothness notions. Then, we construct a limiter as a linear combination of the members of parameter‐dependent families of basis functions, and we explore the possibility to optimize the parameters in interesting model situations to find a corresponding optimal limiter. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservations laws

We introduce a simple model of two conservation laws which is strictly hyperbolic except for a degenerate parabolic line in the state space. Besides classical shock waves, it also exhibits overcompressive, marginal overcompressive, and marginal undercompressive shock waves. Our purpose is to study the behavior of the corresponding viscous waves, in particular the manner in which these waves are stable. There are several basic differences between classical shock waves and other types of shock waves. A perturbation of an overcompressive shock wave gives rise to a new wave. Monotone marginal overcompressive waves behave distinctly from the nonmonotone ones. Analytical techniques used in our study include characteristic-energy and weighted-energy methods, and nonlinear superposition through time-invariants. Although we carry out our analysis for a simple model, the general phenomena would hold for overcompressive waves which occur in other physical models.     

Relaxation WENO schemes for multidimensional hyperbolic systems of conservation laws

We present a class of high‐order weighted essentially nonoscillatory (WENO) reconstructions based on relaxation approximation of hyperbolic systems of conservation laws. The main advantage of combining the WENO schemes with relaxation approximation is the fact that the presented schemes avoid solution of the Riemann problems due to the relaxation approach and high‐resolution is obtained by applying the WENO approach. The emphasis is on a fifth‐order scheme and its performance for solving a wide class of systems of conservation laws. To show the effectiveness of these methods, we present numerical results for different test problems on multidimensional hyperbolic systems of conservation laws. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

BV-estimates of Lax–Friedrichs' scheme for hyperbolic conservation laws with relaxation

In this paper, we will give BV-estimates of Lax–Friedrichs' scheme for a simple hyperbolic system of conservation laws with relaxation and get the global existence and uniqueness of BV-solution by the BV-estimates above. Furthermore, our results show that the solution converge towards the solution of an equilibrium model as the relaxation time ε>0 tends to zero provided sub-characteristic condition holds. Copyright © 2007 John Wiley & Sons, Ltd.     

Global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary

This work is a continuation of our previous work, in the present paper we study the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws with non-linear boundary conditions in the half space . Under the assumption that each characteristic with positive velocity is linearly degenerate, we prove the existence and uniqueness of global weakly discontinuous solution u=u(t,x) with small amplitude, and this solution possesses a global structure similar to that of the self-similar solution of the corresponding Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines, particularly to the system describing the motion of the relativistic string in Minkowski space R1+n, are also given.     

Multiresolution algorithms for the numerical solution of hyperbolic conservation laws

Given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one-dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner. To do so we consider the grid-averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest, and introduce an equivalent multiresolution representation. The multiresolution representation of the numerical solution consists of its grid-averages for the coarsest grid and the set of errors in predicting the grid-averages of each level of resolution in this hierarchy from those of the next coarser one. Once the numerical solution is resolved to our satisfaction in a certain locality of some grid, then the prediction errors there are small for this particular grid and all finer ones; this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance. Therefore instead of computing the time-evolution of the numerical solution on the given grid we compute the time-evolution of its compressed multiresolution representation. Algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid. At each step of refinement we add the values of the numerical flux at the center of the coarser cells. The information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well-resolved. When this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid. The computational efficiency of this multiresolution algorithm is proportional to the rate of data compression (for a prescribed level of tolerance) that can be achieved for the numerical solution of the given scheme.     

Construction of solutions for mixed hyperbolic elliptic Riemann initial value system of conservation laws

In this paper, we apply Adomian decomposition method (ADM) to develop a fast and accurate algorithm for systems of conservation laws of mixed hyperbolic elliptic type. The solutions of our model equations are calculated in the form of convergent power series with easily computable components. The results obtained are compared with our Modification of Adomian decomposition method (MADM) Az-Zo’bi and Al-Khaled (2010) [1]. The study outlines the significant features of the two methods. With application to van der Waals system, we obtain the stability of the approximate solutions graphically when the system changes type with more efficiency of the MADM.     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

Artifical and physical viscosity solutions for a hyperbolic conservation system

In this paper, a new maximum principle is introduced to study positive lower bounds of the density both for the artificial viscosity solutions and for the physical viscosity solutions of a 'hyperbolic conservationsystem derived from the Broadwell model and the global existence of theseviscosity solutions is obtained. We give some simple numerical results fordiscontinuous initial data.