Stratified solutions for systems of conservation laws

We study a class of weak solutions to hyperbolic systems of conservation (balance) laws in one space dimension, called stratified solutions. These solutions are bounded and ``regular' in the direction of a linearly degenerate characteristic field of the system, but not in other directions. In particular, they are not required to have finite total variation. We prove some results of local existence and uniqueness.

     

Non‐selfsimilar global solutions to a two‐dimensional system of conservation laws

This paper considers the two‐dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non‐selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non‐selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Delta shock waves with Dirac delta function in both components for systems of conservation laws

We study a class of non-strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype. The Riemann problems are constructively solved. The Riemann solutions include two kinds of interesting structures. One involves a cavitation where both state variables tend to zero forming a singularity, the other is a delta shock wave in which both state variables contain Dirac delta function simultaneously. The generalized Rankine–Hugoniot relation and entropy condition are proposed to solve the delta shock wave. Moreover, with the limiting viscosity approach, we show all of the existence, uniqueness and stability of solution involving the delta shock wave. The generalized Rankine–Hugoniot relation is also confirmed. Then our theory is successfully applied to two typical systems including the geometric optics equations. Finally, we present the numerical results coinciding with the theoretical analysis.     

Convergence of relaxation schemes for conservation laws

We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the L^∞ and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated.     

Multi‐dimensional conservation laws and integrable systems

In this paper, we introduce a new property of two‐dimensional integrable hydrodynamic chains—existence of infinitely many local three‐dimensional conservation laws for pairs of integrable two‐dimensional commuting flows. Infinitely many local three‐dimensional conservation laws for the Benney commuting hydrodynamic chains are constructed. As a by‐product, we established a new method for computation of local conservation laws for three‐dimensional integrable systems. The Mikhalëv equation and the dispersionless limit of the Kadomtsev‐Petviashvili equation are investigated. All known local and infinitely many new quasilocal three‐dimensional conservation laws are presented. Also four‐dimensional conservation laws are considered for couples of three‐dimensional integrable quasilinear systems and for triplets of corresponding hydrodynamic chains.     

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     

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.     

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.     

New conservation laws of some third-order systems of pdes arising from higher-order multipliers

In this paper, we study conservation laws for some third-order systems of pdes, viz., some versions of the Boussinesq equations and the BBM equation. It is shown that new and interesting conserved quantities arise from ‘multipliers’ that are of order greater than one in derivatives of the dependent variables. Furthermore, the invariance properties of the conserved flows with respect to the Lie point symmetry generators are investigated via the symmetry action on the multipliers.     

高维非线性守恒律方程组

本文根据高维非线性守恒律方程组的研究历程将这一领域的研究大体分为四个阶段: 局部经典解、具扇状波结构弱解、具花状波结构弱解、整体解与混合型方程. 本文据此线索回顾与介绍多年来在该领域所获得的主要成果与进展, 并提出今后所面临的一些未解决的重要问题及困难.     

On the linearization of systems of conservation laws for fluids at a material contact discontinuity

In order to investigate the linearized stability or instability of compressible flows, as it occurs for instance in Rayleigh–Taylor or Kelvin–Helmholtz instabilities, we consider the linearization at a material discontinuity of a flow modeled by a multidimensional nonlinear hyperbolic system of conservation laws. Restricting ourselves to the plane-symmetric case, the basic solution is thus a one-dimensional contact discontinuity and the normal modes of pertubations are solutions of the resulting linearized hyperbolic system with discontinuous nonconstant coefficients and source terms. While in Eulerian coordinates, the linearized Cauchy problem has no solution in the class of functions, we prove that for a large class of systems of conservation laws written in Lagrangian coordinates and including the Euler and the ideal M.H.D. systems, there exists a unique function solution of the problem that we construct by the method of characteristics.     

Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions

In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time‐asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter ε tends to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

Periodic solutions of conservation laws constructed through Glimm scheme

We present a periodic version of the Glimm scheme applicable to special classes of systems for which a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973) is available. For these special classes of systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in , for Bakhvalov's class, and in , in the case of DiPerna's class. It may also happen that the system is in Bakhvalov's class only at a neighboorhood of a constant state, as it was proved for the isentropic gas dynamics by DiPerna (1973), in which case the initial data is taken in with , for some constant which is for the isentropic gas dynamics systems. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, , are uniformly bounded in , for all 0$">, where is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen and Frid in (1999) combined with a compactness theorem of DiPerna in (1983). The question about the decay of Nishida's solution was proposed by Glimm and Lax in (1970) and has remained open since then. The classes considered include the -systems with , , , which, for , model isentropic gas dynamics in Lagrangian coordinates.

     

Local conservation laws,symmetries, and exact solutions for a Kudryashov‐Sinelshchikov equation

In this paper, we consider a Kudryashov‐Sinelshchikov equation that describes pressure waves in a mixture of a liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer between liquid and gas bubbles. We show that this equation is rich in conservation laws. These conservation laws have been found by using the direct method of the multipliers. We apply the Lie group method to derive the symmetries of this equation. Then, by using the optimal system of 1‐dimensional subalgebras we reduce the equation to ordinary differential equations. Finally, some exact wave solutions are obtained by applying the simplest equation method.     

Explicit generalized solutions to a system of conservation laws

This paper studies a special 3 by 3 system of conservation laws which cannot be solved in the classical distributional sense. By adding a viscosity term and writing the system in the form of a matrix Burgers equation an explicit formula is found for the solution of the pure initial value problem. These regularized solutions are used to construct solutions for the conservation laws with initial conditions, in the algebra of generalized functions of Colombeau. Special cases of this system were studied previously by many authors.     

Hyperbolic conservation laws with nonlinear diffusion and nonlinear dispersion

We study scalar conservation laws with nonlinear diffusion and nonlinear dispersion terms (any ??1), the flux function f(u) being mth order growth at infinity. It is shown that if ε, δ=δ(ε) tend to 0, then the sequence {uε} of the smooth solutions converges to the unique entropy solution u∈L^∞(0,T_∗;Lq(R)) to the conservation law ut+fx(u)=0 in . The proof relies on the methods of compensated compactness, Young measures and entropy measure-valued solutions. Some new a priori estimates are carried out. In particular, our result includes the convergence result by Schonbek when b(λ)=λ, ?=1 and LeFloch and Natalini when ?=1.     

Local error analysis for approximate solutions of hyperbolic conservation laws

We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted intoL _loc estimates, following theLip convergence theory developed by Tadmor et al. Comparisons between the local truncation error and theL _loc -error show remarkably similar behavior. Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms in [Karni, Kurganov and Petrova, J. Comput. Phys. 178 (2002) 323–341].     

Monotone first‐order weighted schemes for scalar conservation laws

In this work, we present a monotone first‐order weighted (FORWE) method for scalar conservation laws using a variational formulation. We prove theoretical properties as consistency, monotonicity, and convergence of the proposed scheme for the one‐dimensional (1D) Cauchy problem. These convergence results are extended to multidimensional scalar conservation laws by a dimensional splitting technique. For the validation of the FORWE method, we consider some standard bench‐mark tests of bidimensional and 1D conservation law equations. Finally, we analyze the accuracy of the method with L¹ and L^∞ error estimates. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013