Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long-time behavior governed by reduced systems that are parabolic in nature. In this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid. We introduce a modification of higher order Godunov methods that possesses the correct asymptotic behavior, allowing the use of coarse grids (large cell Peclet numbers). The idea is to build into the numerical scheme the asymptotic balances that lead to this behavior. Numerical experiments on 2 \Theta 2 systems verify our analysis. 1 E-mail address: jin@math.gatech.edu 2 E-mail address: lvrmr@math.arizona.edu Typeset by A M S-T E X 2 1. Introduction Hyperbolic systems of partial differential equations that arise in applications ofter have re...

Underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well-resolved. We design a second order Runge-Kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit, thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved. Key words. Hyperbolic conservation laws with stiff relaxation, shock capturing difference method, Runge-Kutta methods, asymptotic limit AMS(MOS) subject classifications. 35L65, 35B40, 65M60 Typeset by A M S-T E X 2 1. Introduction Hyperbolic systems with relaxations occur in the study of a variety of physical phenomena, for example in linear and nonlinear waves [42,36], in relaxing gas flow with thermal and chemical nonequilibrium [41,9], in kinetic theory of ra...

In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible flow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit second-order Godunov algorithm followed by a corrector applied to cells at the boundary. The discretization near the fluid-body interface is based on a volume-of-fluid approach with a redistribution procedure to maintain conservation while avoiding time step restrictions arising from small cells where the boundary intersects the mesh. The single grid Cartesian mesh integration scheme is coupled to a conservative adaptive mesh refinement algorithm that selectively refines regions of the computational grid to achieve a desired level of accuracy. Examples showing the results of the combined Cartesian grid integration/adaptive mesh refinement algorithm for both two- and three-dimensional flows are presented. (This page intent...

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...

. A class of numerical schemes for nonlinear kinetic equations of Boltzmann type is described. Following Wild's approach, the solution is represented as a power series with parameter depending exponentially on the Knudsen number. This permits us to derive accurate and stable time discretizations for all ranges of the mean free path. These schemes preserve the main physical properties: positivity, conservation of mass, momentum, and energy. Moreover, for some particular models, the entropy property is also shown to hold. Key words. Boltzmann equation, fluid dynamic limit, Wild sum AMS subject classifications. 35L65, 65C20, 76P05, 82C40 PII. S0036142995287768 1. Introduction. Numerical resolution methods for the Boltzmann equation play an important role in practical and theoretical analysis of the time evolution of a rarefied gas. The widely used and best-known of these methods is the direct simulation Monte Carlo method due to Bird [4]. After Bird's algorithm, more sophisticated meth...

We analyze a simple system of conservation laws with a strong relaxation term. Wellposedness of the Cauchy problem, in the framework of BV-solutions, is proved. Furthermore, we prove that the solutions converge towards the solution of an equilibrium model as the relaxation time ffi ? 0 tends to zero. Finally, we show that the difference between an equilibrium solution (ffi = 0) and a non-equilibrium solution (ffi ? 0), measured in L 1 , is bounded by O(ffi 1=3 ).

Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small-relaxation limit governed by reduced systems of parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved. We develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit, which thus are able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type.

The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a non-conservative reformulation of the zero-order terms of the right-hand-side of the equations. In this context, we decided to use the results of DalMaso, LeFloch and Murat [9] about non-conservative products, and the generalized Roe matrixes introduced by Toumi [36] to derive a first-order linearized well-balanced scheme in the sense of Greenberg and LeRoux [19]. As a main feature, this approach is able to preserve the right asymptotic behaviour of the original inhomogeneous system [31], which is not a obvious property [6]. Numerical results for the Euler equations are shown to handle correctly these equilibria in various situations. Key words: conservation laws, source terms. nonconservative products, balanced scheme. AMS subjects classification: 65M06, 76N15. 1 Current adress: Foundation for Research and Technology Hel...

this paper we shall only consider systems with relaxation. Although very efficient and accurate methods have been developped for both hyperbolic systems and systems of ordinary differential equations, many numerical schemes for hyperbolic systems with relaxation are unsatisfactory and the main difficulty arises from the need to handle very different relaxation times with the same scheme. For instance solid particles are usually added in rocket engines in order to damp the combustion instabilities. The particles burn inside the rocket so that the stiffness of the drag terms range from nonstiff to very stiff. On the other hand the computation of an initial value problem for an hyperbolic system with relaxation also involves a wide range of stiffness of the source terms: if the initial data is away from equilibrium, there is a boundary layer in time of order # after which the solution is close to equilibrium. During a time interval of order # the relaxation terms are thus stiff while they become nonstiff after a time of order # . The chalenge is thus to construct a numerical scheme that Numerische Mathematik Electronic Edition page 144 of Numer. Math. 77: 143--185 (1997) A Riemann solver for hyperbolic systems with relaxation 145 may handle any stiffness and whose computational cost is of the same order as the cost of usual methods such as the Strang splitting for instance: in order that the computational cost of the method be of the same order as the cost of usual methods for hyperbolic systems of conservation laws, we want to chose the time step only on the CFL condition relative to the convection terms and the source terms should be underresolved in the stiff case. The construction of numerical schemes for hyperbolic systems with relaxation has attracted a lot of atte...