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85
SemiDiscrete CentralUpwind Schemes for Hyperbolic Conservation Laws and HamiltonJacobi Equations
 SIAM J. Sci. Comput
, 2000
"... We introduce new Godunovtype semidiscrete central schemes for hyperbolic systems of conservation laws and HamiltonJacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation, and can be viewed as a generalization of the schemes from [26, 24 ..."
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Cited by 45 (3 self)
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We introduce new Godunovtype semidiscrete central schemes for hyperbolic systems of conservation laws and HamiltonJacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation, and can be viewed as a generalization of the schemes from [26, 24, 25] and [27]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the onesided local speeds. This is the reason why we call them centralupwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the HamiltonJacobi equations with convex and nonconvex Hamiltoni...
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.
Compact central WENO schemes for multidimensional conservation laws
 SIAM J. Sci. Comput
, 2000
"... We present new third and fifthorder Godunovtype central schemes for approximating solutions of the HamiltonJacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greate ..."
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Cited by 30 (8 self)
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We present new third and fifthorder Godunovtype central schemes for approximating solutions of the HamiltonJacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greater than two. In two space dimensions we present two versions for the thirdorder scheme: one scheme that is based on a genuinely twodimensional Central WENO reconstruction, and another scheme that is based on a simpler dimensionbydimension reconstruction. The simpler dimensionbydimension variant is then extended to a multidimensional fifthorder scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes. Key words. HamiltonJacobi equations, central schemes, high order, WENO, CWENO.
D.: A thirdorder semidiscrete central scheme for conservation laws and convectiondiffusion equations
 SIAM J. Sci. Comput
, 2000
"... We present a new thirdorder, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convectiondiffusion equations, and related problems. Our method is a highorder extension of the recently proposed secondorder, semidiscrete method ..."
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Cited by 26 (2 self)
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We present a new thirdorder, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convectiondiffusion equations, and related problems. Our method is a highorder extension of the recently proposed secondorder, semidiscrete method in [16]. The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cellaverages. We demonstrate our results, by focusing on the new thirdorder CWENO reconstruction presented in [21]. The numerical results we present, show the desired accuracy, high resolution and robustness of our method. Key words. Hyperbolic systems, convectiondiffusion equations, central difference schemes, highorder accuracy, nonoscillatory schemes, WENO reconstruction. AMS(MOS) subject classification. Primary 65M10; secondary 65M05.
A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes
 J. Comput. Phys
, 2001
"... We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of appro ..."
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Cited by 23 (5 self)
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We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.
Solution of twodimensional Riemann problems for gas dynamics without Riemann problem solvers
 Numer. Methods Partial Differential Equations
, 2002
"... We report here on our numerical study of the twodimensional Riemann problem for the compressible Euler equations. Compared with the relatively simple 1D configurations, the 2D case consists of a plethora of geometric wave patterns that pose a computational challenge for highresolution methods. T ..."
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Cited by 22 (2 self)
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We report here on our numerical study of the twodimensional Riemann problem for the compressible Euler equations. Compared with the relatively simple 1D configurations, the 2D case consists of a plethora of geometric wave patterns that pose a computational challenge for highresolution methods. The main feature in the present computations of these 2D waves is the use of the Riemannsolversfree central schemes presented by Kurganov et al. This family of central schemes avoids the intricate and timeconsuming computation of the eigensystem of the problem and hence offers a considerably simpler alternative to upwind methods. The numerical results illustrate that despite their simplicity, the central schemes are able to recover with comparable high resolution, the various features observed in the earlier,
Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients
 M2AN Math. Model. Numer. Anal
"... Abstract. We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a “rough ” coefficient function k(x). We show that the EngquistOsher (and hence all monotone) finite difference approximations ..."
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Cited by 19 (8 self)
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Abstract. We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a “rough ” coefficient function k(x). We show that the EngquistOsher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k ′ is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convectiondiffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general L p compactness criterion.
New HighResolution SemiDiscrete Central Schemes for HamiltonJacobi Equations
 JCP
, 2000
"... ..."
Strongly degenerate parabolichyperbolic systems modeling polydisperse sedimentation with compression
 SIAM J. APPL. MATH
, 2003
"... We show how existingmodels for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions formingcompressible sediments (“sedimentation w ..."
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Cited by 15 (7 self)
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We show how existingmodels for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions formingcompressible sediments (“sedimentation with compression” or “sedimentationconsolidation process”). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convectiondiffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolichyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov–Tadmor central difference scheme for convectiondiffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.
Central Schemes For Balance Laws Of Relaxation Type
 SIAM J. NUMER. ANAL
, 2000
"... Several models in mathematical physics are described by quasilinear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodyna ..."
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Cited by 14 (2 self)
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Several models in mathematical physics are described by quasilinear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodynamics are presented. The numerical methods are a generalization of the NessyahuTadmor scheme to the nonhomogeneous case byincluding the cell averages of the production terms in the discrete balance equations. A second order scheme uniformlyaccurate in the relaxation parameter is derived and its properties analyzed. Numerical tests confirm the accuracy and robustness of the scheme.