Results 1  10
of
48
Uniformly accurate schemes for hyperbolic systems with relaxations
 SIAM J. Numer. Anal
, 1997
"... Abstract. We develop highresolution shockcapturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order1 to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underreso ..."
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Cited by 61 (21 self)
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Abstract. We develop highresolution shockcapturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order1 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 cannot 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 secondorder scheme that works effectively, with a fixed spatial and temporal discretization, for all ranges of the mean free path. Formal uniform consistency proof for a firstorder scheme and numerical convergence proof for the secondorder scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivated by the reentry problem in hypersonic computations.
Numerical Schemes For Hyperbolic Conservation Laws With Stiff Relaxation Terms
 J. Comput. Phys
, 1996
"... Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a longtime 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 meth ..."
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Cited by 57 (11 self)
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Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a longtime 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 Email address: jin@math.gatech.edu 2 Email address: lvrmr@math.arizona.edu Typeset by A M ST E X 2 1. Introduction Hyperbolic systems of partial differential equations that arise in applications ofter have re...
RungeKutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms
 J. Comput. Phys
, 1995
"... 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 wellresolved. We design a second order RungeKutta type splitting method that posse ..."
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Cited by 48 (14 self)
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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 wellresolved. We design a second order RungeKutta 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, RungeKutta methods, asymptotic limit AMS(MOS) subject classifications. 35L65, 35B40, 65M60 Typeset by A M ST 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...
An Adaptive Cartesian Grid Method For Unsteady Compressible Flow In Irregular Regions
 J. Comput. Phys
, 1993
"... 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 secondorder Godunov algorithm fo ..."
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Cited by 48 (14 self)
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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 secondorder Godunov algorithm followed by a corrector applied to cells at the boundary. The discretization near the fluidbody interface is based on a volumeoffluid 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 threedimensional flows are presented. (This page intent...
Relaxation Schemes For Nonlinear Kinetic Equations
 SIAM J. Numer. Anal
, 1997
"... . 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 ..."
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Cited by 37 (15 self)
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. 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 bestknown of these methods is the direct simulation Monte Carlo method due to Bird [4]. After Bird's algorithm, more sophisticated meth...
Discrete Kinetic Schemes For Multidimensional Systems Of Conservation Laws
 SIAM J. Numer. Anal
, 2000
"... 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 ..."
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Cited by 33 (11 self)
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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.
Convergence of Relaxation Schemes for Conservation Laws
, 1996
"... . 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 se ..."
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Cited by 32 (10 self)
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. 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 ST E X 2 Convergence of R...
On The Rate Of Convergence To Equilibrium For A System Of Conservation Laws Including A Relaxation Term
 SIAM J. Math. Anal
, 1994
"... We analyze a simple system of conservation laws with a strong relaxation term. Wellposedness of the Cauchy problem, in the framework of BVsolutions, 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 ..."
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Cited by 25 (4 self)
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We analyze a simple system of conservation laws with a strong relaxation term. Wellposedness of the Cauchy problem, in the framework of BVsolutions, 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 nonequilibrium solution (ffi ? 0), measured in L 1 , is bounded by O(ffi 1=3 ).
Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation
 SIAM J. Numer. Anal
, 1997
"... Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a smallrelaxation 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 ..."
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Cited by 25 (11 self)
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Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a smallrelaxation 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.
A WellBalanced Scheme Using NonConservative Products Designed for Hyperbolic Systems of Conservation Laws With Source Terms
, 2001
"... 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 nonconservative reformulation of the zeroorder terms of the righthandside of the equations. In this context, we decided to use the ..."
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Cited by 21 (3 self)
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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 nonconservative reformulation of the zeroorder terms of the righthandside of the equations. In this context, we decided to use the results of DalMaso, LeFloch and Murat [9] about nonconservative products, and the generalized Roe matrixes introduced by Toumi [36] to derive a firstorder linearized wellbalanced 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...