We construct, analyze, and implement a new nonoscillatory high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewise-linear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a second-order, nonoscillatory central scheme, a natural extension of the one-dimensional second-order central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408--448]. As in the one-dimensional case, the main feature of our two-dimensional scheme is simplicity. In particular, this central scheme does not require the intricate and time-consuming (approximate) Riemann solvers which are essential for the high-resolution upwind schemes; in fact, even the com...

We study the Cauchy problem for 2\Theta2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to zero. This research was strongly motivated by the recent numerical investigations of S. Jin and Z. Xin on the relaxation schemes for conservation laws. 1. Introduction In this paper we are interested to the relaxation behaviour of the following system of hyperbolic conservation laws with a singular perturbation source (1.1) ae @ t u + @ x v = 0 ; @ t v + @ x oe(u) = \Gamma 1 " (v \Gamma f(u)) (" ? 0); for (x; t) 2 IR \Theta (0; 1). Here oe, f are some given smooth functions such that oe 0 (u) ( ? 0), f(0) = 0. The system (1.1) is equivalent to the one-dimensional perturbed wave equation (1.2) @ tt w \Gamma @ x oe(@ x...

We develop high resolution shock capturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order one to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. Usually one can not decouple the problem into separate regimes and handle different regimes with different methods. Thus it is important to have a scheme that works uniformly with respect to the relaxation time. Using the Broadwell model of the nonlinear Boltzmann equation we develop a second order scheme that works effectively, with a fixed spatial and temporal discretization, for all range of mean free path. Formal uniform consistency proof for a first order scheme, and numerical convergence proof for the second order scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivate...

. A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: #1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given cell averages; and #2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a thirdorder central scheme, an extension along the lines of the second-order central scheme of Nessyahu and Tadmor [NT]. The scalar scheme is non-oscillatory (and hence -- convergent), in the sense that it does not increase the number of initial extrema (--as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the highresolution content of...

this article, entropy weak solutions of (1.1) are constructed as the singular limit of associated relaxation systems. The relaxation models can be loosely interpreted as (and their selection was motivated by) discrete velocity kinetic equations. The relaxation parameter plays the role of the mean free path and the system models the macroscopic conservation law at a "mesoscopic" level. In that sense they are a discrete velocity analogue of the kinetic equations introduced by Perthame and Tadmor [24], Lions, Perthame and Tadmor [19], which also describe entropy solutions of (1.1) as the mean free path tends to zero and provide a kinetic formulation for the scalar conservation law. Furthermore, we may go back to the "microscopic" level and build a stochastic interacting particle system, which: (a) in mesoscopic scales approximates the relaxation model and (b) in the fluid-dynamic limit converges to the entropy solution of a scalar conservation law. This issue was addressed by Perthame and Pulvirenti [23] for interacting particle systems corresponding to the kinetic formulation of conservation laws in [19], and is undertaken in a companion article [14] for interacting particle systems induced by the relaxation schemes proposed here. These results as well as part of the work in the present article are announced in the note [13]. The presence of relaxation mechanisms is widespread in both the continuum mechanics as well as the kinetic theory contexts. Relaxation is known to provide a subtle "dissipative" mechanism for discontinuities against the destabilizing effect of nonlinear response [20], as well as a damping effect on oscillations [3] - at least when assisted by nonlinear response. We refer to Chen, Levermore and Liu [3] for a discussion of a general mathematical framew...

We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whitham's stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.

. 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 1 and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated. Key words and phrases: Relaxation schemes, conservation laws, shock waves, entropy conditions, hyperbolic singular perturbations. 1. Introduction In this paper we investigate a new class of numerical schemes, which are based on the local relaxation approximation of conservation laws. Consider the initial value problem (1.1) @ t u + @ x f(u) = 0 ; (1.2) u(x; 0) = u 0 (x) for (x; t) 2 IR \Theta (0; 1). Here f is a given (say C 1 ) smooth function such that f(0) = f 0 (0) = 0. Typeset by A M S-T E X 2 Convergence of R...

We present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities. Under a suitable multidimensional generalization of the Whitham relaxation subcharacteristic condition, we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem. 1 Introduction Consider a multidimensional conservation law. Let u : R d \Theta R+ ! R be a global weak solution to the Cauchy problem @ t u + d X j=1 @ x j A j (u) = 0 ; (x; t) 2 R d \Theta R+ ; (1.1) u(x; 0) = u 0 (x) ; x 2 R d ; (1.2) where A = (A 1 ; : : : ; A d ) 2 (Lip loc (R)) d and u 0 2 L 1 (R d ). In this paper we shall restrict our consideration to the entropy solutions of (1.1)--(1.2) in the sense of Kruzkov, which are known to be unique and having good mathematical properties, see [Kr] and Section 3 below. Consider also the following class of discret...

During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical tools which are used in the development of convergence theories for these algorithms. These include classical compactness arguments (based on BV a priori estimates), the use of compensated compactness arguments (based on H^-1-compact entropy production), measure valued solutions (measured by their negative entropy production), and finally, we highlight the most recent addition to this bag of analytical tools -- the use of averaging lemmas which yield new compactness and regularity results for nonlinear conservation laws and related equations. We demonstrate how these analytical tools are used in the convergence analysis of approximate solutions for hyperbolic conservation laws and related equations. Our discussion includes examples of Total Variation Diminishing (TVD) finite-difference schemes; error estimates derived from the one-sided stability of Godunov-type methods for convex conservation laws (and their multidimensional analogue -- viscosity solutions of demi-concave Hamilton-Jacobi equations); we outline, in the one-dimensional case, the convergence proof of finite-element streamline-diffusion and spectral viscosity schemes based on the div-curl lemma; we also address the questions of convergence and error estimates for multidimensional finite-volume schemes on non-rectangular grids; and finally, we indicate the convergence of approximate solutions with underlying kinetic formulation, e.g., finite-volume and relaxation schemes, once their regularizing effect is quantified in terms of the averaging lemma.

Abstract. In recent work, the second author and various collaborators have shown using Evans function/refined semigroup techniques that, under very general circumstances, the problems of determining one- or multi-dimensional nonlinear stability of a smooth shock profile may be reduced to that of determining spectral stability of the corresponding linearized operator about the wave. It is expected that this condition should in general be analytically verifiable in the case of small amplitude profiles, but this has so far been shown only on a case-by-case basis using clever (and difficult to generalize) energy estimates. Here, we describe how the same set of Evans function tools that were used to accomplish the original reduction can be used to show also small-amplitude spectral stability by a direct and readily generalizable procedure. This approach both recovers the results obtained by energy methods, and yields new results not previously obtainable. In particular, we establish one-dimensional stability of small amplitude relaxation profiles, completing the Evans function program set out in Mascia&Zumbrun [MZ.1]. Multidimensional stability of small amplitude viscous profiles will be addressed in a companion paper [PZ], completing the program of Zumbrun [Z.3]. Section