Existence and uniqueness of solutions for some hyperbolic systems of conservation laws

We study the Cauchy problem for systems of conservation laws which belong to the Temple class. The compensated-compactness theory is used to prove existence of solutions. Some uniqueness results are established by means of Holmgren's principle.     

Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws

We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave.     

The method of quasidecoupling for discontinuous solutions to conservation laws

Communicated by C. M. Dafermos     

Non-observable chaos in piecewise smooth systems

In the present paper, we discuss bifurcations of chaotic attractors in piecewise smooth one-dimensional maps with a high number of switching manifolds. As an example, we consider models of DC/AC power electronic converters (inverters). We demonstrate that chaotic attractors in the considered class of models may contain parts of a very low density, which are unlikely to be observed, neither in physical experiments nor in numerical simulations. We explain how the usual bifurcations of chaotic attractors (merging, expansion and final bifurcations) in piecewise smooth maps with a high number of switching manifolds occur in a specific way, involving low-density parts of attractors, and how this leads to an unusual shape of the bifurcation diagrams.     

Decay of solutions of hyperbolic systems of conservation laws with a convex extension

Archive for Rational Mechanics and Analysis -     

Application of conservation laws to Dirichlet problem for elliptic quasilinear systems

In the present work we show the possibility of using of conservation laws to solve the Dirichlet problem for elliptic quasilinear systems. As a result the integral representation of solution is obtained. For the system of filtration of aerated fluid in porous medium and for system of elastic–plastic torsion of prismatic rods corresponding conservation laws are calculated in explicit form and the Dirichlet problems are solved.     

Methods for extending high-resolution schemes to non-linear systems of hyperbolic conservation laws

In extending high-resolution methods from the scalar case to systems of equations there are a number of options available. These options include working with either conservative or primitive variables, characteristic decomposition, two-step methods, or component-wise extension. In this paper, several of these options are presented and compared in terms of economy and solution accuracy. The characteristic extension with either conservative or primitive variables produces excellent results with all the problems solved. Using primitive variables, the two-step formulation produces high-quality results in a more economical manner. This method can also be extended to multiple dimensions without resorting to dimensional splitting. Proper selection of limiters is also important in non-characteristic extension to systems.     

Adaptive multi-resolution central-upwind schemes for systems of conservation laws

In this article, we develop an adaptive scheme for solving systems of hyperbolic conservation laws. In this scheme nonlinear shock and linear contact waves will be treated differently. The proposed scheme uses the Kurganov central-upwind scheme. Fourth-order non-oscillatory reconstruction is employed near shock only while the unlimited fifth-order reconstruction is used for smooth regions and linear contact waves. To distinguish between the smooth parts and discontinuities, we use an efficient adaptive multi-resolution technique. The advantages of the scheme are high resolution and computationally efficient since limiters are used only for shocks. Numerical experiments with one- and two-dimensional problems are presented which show the robustness of the proposed scheme.     

Integrating factors and conservation laws for nonconservative dynamical systems

A general approach to the construction of conservation laws for classical nonconservative dynamical systems is presented. The conservation laws are constructed by finding corresponding integrating factors for the equations of motion. Necessary conditions for existence of the conservation laws are studied in detail. A connection between an a priori known conservation law and the corresponding integrating factors is established. The theory is applied to two particular problems.     

Topology of elementary waves for mixed-type systems of conservation laws

We construct explicitly the fundamental wave manifold for systems of two conservation laws with quadratic flux functions. We describe the shock foliation for this manifold, as well as the singular set of the foliation. We subdivide the manifold into regions where the shock curves form trivial foliations. Sonic surfaces are identified as well. We establish the stability of shock curves underC ³ perturbations of the flux functions in the Whitney topology.In memoriam of Jean Martinet.     

Nonlinear stability of discrete shocks for systems of conservation laws

In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m×m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the L ^p-norm for all p 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in L ^p (P 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.     

Complete systems of conservation laws for two-layer shallow water models

A well-posedness criterion for a complete system of conservation laws is proposed that assumes maximum compatibility of the convexity domain of the closing conservation law with the domain of hyperbolicity of the model used. This criterion is used to obtain well-posed complete systems of conservation laws for the models of two-layer shallow water with a free-surface (model I) and with a rigid lid (model II). Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 23–32, September–October, 1999.     

Measure-valued solutions of scalar conservation laws with boundary conditions

We define a solution concept for measure-valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions. This result may be used to prove convergence, towards the unique solution, for approximate solutions which are uniformly bounded in L , weakly consistent with certain entropy inequalities and strongly consistent with the initial condition, i.e. without using derivative estimates. As an example convergence of a finite element method is demonstrated.