A quantitative compactness estimate for scalar conservation laws

In the case of a scalar conservation law with convex flux in space dimension one, P. D. Lax proved [Comm. Pure and Appl. Math. 7 (1954)] that the semigroup defining the entropy solution is compact in L for each positive time. The present note gives an estimate of the ?‐entropy in L of the set of entropy solutions at time t > 0 whose initial data run through a bounded set in L¹. © 2005 Wiley Periodicals, Inc.     

Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

Summary. In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law with initial condition . The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semi-implicit in the stiff case) and the entropy solution. The order of these estimates is in space-time -norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case. Received October 21, 1999 / Published online February 5, 2001     

Local variational problems and conservation laws

We investigate globality properties of conserved currents associated with local variational problems admitting global Euler–Lagrange morphisms. We show that the obstruction to the existence of a global conserved current is the difference of two conceptually independent cohomology classes: one coming from using the symmetries of the Euler–Lagrange morphism and the other from the system of local Noether currents.     

Parallel adaptive hp-refinement techniques for conservation laws

We describe an adaptive hp-refinement local finite element procedure for the parallel solution of hyperbolic systems of conservation laws on rectangular domains. The local finite element procedure utilizes spaces of piecewise-continuous polynomials of arbitrary degree and coordinated explicit Runge-Kutta temporal integration. A solution limiting procedure produces monotonic solutions near discontinuities while maintaining high-order accuracy near smooth extrema. A modified tiling procedure maintains processor load balance on parallel, distributed-memory MIMD computers by migrating finite elements between processors in overlapping neighborhoods to produce locally balanced computations. Grids are stored in tree data structures, with finer grids being offspring of coarser ones. Within each grid, AVL trees simplify the transfer of information between neighboring processors and the insertion and deletion of elements as they migrate between processors. Computations involving Burgers' and Euler's equations of inviscid flow demonstrate the effectiveness of the hp-refinement and balancing procedures relative to non-balanced adaptive and balanced non-adaptive procedures.     

Two new moving boundary problems for scalar conservation laws

The application of the theory of scalar conservation laws to semiconductor device fabrication is described. This application is the source of a Stefan problem and another moving boundary problem for a class of such equations. The analogue of the Riemann problem for these problems is analyzed and solved. Conditions on the boundary values that characterize physically correct solutions are derived.     

An optimal order estimate for nonlinear hyperbolic conservation laws in two variables

The standard Galerkin method is studied for nonlinear hyperbolic conservation laws in two space variables. An optimal L²-estimate is proved on arbitrary rectangular meshes for smooth periodic solutions.     

A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems

Summary. We construct a new third-order semi-discrete genuinely multidimensional central scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is independent of , allows us to pass to a limit as . This results in a particularly simple genuinely multidimensional semi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, proposed in [29]. Then, we generalize it for the computation of the two-dimensional numerical fluxes. Our scheme enjoys the main advantage of the Godunov-type central schemes –simplicity, namely it does not employ Riemann solvers and characteristic decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our scheme. Received February 7, 2000 / Published online December 19, 2000     

Rate conservation laws: A survey

We survey the rate conservation law, RCL for short, arising in queues and related stochastic models. RCL was recognized as one of the fundamental principles to get relationships between time and embedded averages such as the extended Little's formulaH=G, but we show that it has other applications. For example, RCL is one of the important techniques for deriving equilibrium equations for stochastic processes. It is shown that the various techniques, including Mecke's formula for a stationary random measure, can be formulated as RCL. For this purpose, we start with a new definition of the rate with respect to a random measure, and generalize RCL by using it. We further introduce the notion of quasi-expectation, which is a certain extension of the ordinary expectation, and derive RCL applicable to the sample average results. It means that the sample average formulas such asH=G can be obtained as the stationary RCL in the quasi-expectation framework. We also survey several extensions of RCL and discuss examples. Throughout the paper, we would like to emphasize how results can be easily obtained by using a simple principle, RCL.     

Semi-definite programming techniques for structured quadratic inverse eigenvalue problems

In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs.     

Uniqueness via the adjoint problems for systems of conservation laws

We prove a result of uniqueness of the entropy weak solution to the Cauchy problem for a class of nonlinear hyperbolic systems of conservation laws that includes in particular the p-system of isentropic gas dynamics. Our result concerns weak solutions satisfying the, as we call it, Wave Entropy Condition, or WEC for short, introduced in this paper. The main feature of this condition is that it concerns both shock waves and rarefaction waves present in a solution. For the proof of uniqueness, we derive an existence result (respectively a uniqueness result) for the backward (respectively forward) adjoint problem associated with the nonlinear system. Our method also applies to obtain results of existence or uniqueness for some linear hyperbolic systems with discontinuous coefficients. © 1993 John Wiley & Sons, Inc.     

Hyperbolic conservation laws on manifolds. An error estimate for finite volume schemes

Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch＇s theory which was originally developed in the Euclidian setting. We extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.     

An optimal error estimate for upwind Finite Volume methods for nonlinear hyperbolic conservation laws

The purpose of this paper is to show that the cell-centered upwind Finite Volume scheme applied to general hyperbolic systems of m conservation laws approximates smooth solutions to the continuous problem at order one in space and time. As it is now well understood, there is a lack of consistency for order one upwind Finite Volume schemes: the truncation error does not tend to zero as the time step and the grid size tend to zero. Here, following our previous papers on scalar equations, we construct a corrector that allows us to prove the expected error estimate for nonlinear systems of equations in one dimension.     

Adjoint IMEX-based schemes for control problems governed by hyperbolic conservation laws

Starting from relaxation schemes for hyperbolic conservation laws we derive continuous and discrete schemes for optimization problems subject to nonlinear, scalar hyperbolic conservation laws. We discuss properties of first- and second-order discrete schemes and show their relations to existing results. In particular, we introduce first and second-order relaxation and relaxed schemes for both adjoint and forward equations. We give numerical results including tracking type problems with non-smooth desired states.     

Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions

We revisit the classical theory of multidimensional scalar conservation laws. We reformulate the notion of the classical Kruzkov entropy solutions and study some new properties as well as the well-posedness of the initial value problem with inhomogeneous fluxes and general initial data. We also consider Dirichlet boundary value problems. We put forward a new and transparent definition for solutions and give a simple proof for their well-posedness in domains with smooth boundaries. Finally, we introduce the notion of saturated solutions and show that it is well-posed.     

Geometry of conservation laws for a class of parabolic PDE's, II: Normal forms for equations with conservation laws

We consider conservation laws for second-order parabolic partial differential equations for one function of three independent variables. An explicit normal form is given for such equations having a nontrivial conservation law. It is shown that any such equation whose space of conservation laws has dimension at least four is locally contact equivalent to a quasi-linear equation. Examples are given of nonlinear equations that have an infinite-dimensional space of conservation laws parameterized (in the sense of Cartan-K?hler) by two arbitrary functions of one variable. Furthermore, it is shown that any equation whose space of conservation laws is larger than this is locally contact equivalent to a linear equation.     

Decay estimate for a new measure of rarefaction waves in general systems of conservation laws

The sharp decay estimate of rarefaction waves in terms of a partial ordering among positive measures is not only interesting in itself but also crucial in the study of convergence rate of vanishing viscosity approximations, cf. Bressan and Yang (2004) [10]. Such an estimate is well established for genuinely nonlinear system of conservation laws, cf. Bressan and Yang (2004) [9]. But similar result is not available for non-genuinely nonlinear system. In this paper, we give a new measure about the rarefaction waves. In addition, a sharp decay estimate of the new measure is given for the cubic nonlinear system of conservation laws.     

A kinetic method for strictly nonlinear conservation laws

Ideas from kinetic theory are used to construct a new solution method for nonlinear conservation laws of the formu ₁+f(u)_x=0. We choose a class of distribution functionsG=G(t, x, ), which are compactly supported with respect to the artificial velocity. This can be done in an optimal way, i.e. so that the-integral of the solution of the linear kinetic equationG _t+G_x=0 solves the nonlinear conservation law exactly.Introducing a time step and variousx-discretisations one easily obtains a variety of numerical schemes. Among them are interesting new methods as well as known upstream schemes, which get a new interpretation and the possibility to incorporate boundary value problems this way.