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166
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.
A Wave Propagation Method for Three Dimensional Hyperbolic Problems
, 1996
"... A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. ..."
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Cited by 23 (5 self)
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A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse direction to model crossderivative terms. Due to proper upwinding, the method is stable for Courant numbers up to one. Several examples using the Euler equations are included.
Derivation of Hyperbolic Models for Chemosensitive Movement
 J. MATH. BIOL
, 2003
"... A ChapmanEnskog expansion is used to derive hyperbolic models for chemosensitive movements as a hydrodynamic limit of a velocityjump process. On the one hand, it connects parabolic and hyperbolic chemotaxis models since the former arise as diffusion limits of a similar velocityjump process. O ..."
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Cited by 22 (2 self)
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A ChapmanEnskog expansion is used to derive hyperbolic models for chemosensitive movements as a hydrodynamic limit of a velocityjump process. On the one hand, it connects parabolic and hyperbolic chemotaxis models since the former arise as diffusion limits of a similar velocityjump process. On the other hand, this approach provides a unified framework which includes previous models obtained by ad hoc methods or methods of moments. Numerical simulations are also performed and are motivated by recent experiments with human endothelial cells on matrigel. Their movements lead to the formation of networks that are interpreted as the beginning of a vasculature. These structures cannot be explained by parabolic models but are recovered by numerical experiments on hyperbolic models.
Contact discontinuity capturing schemes for linear advection and compressible gas dynamics
 J. Sci. Comput
, 2002
"... Abstract We present a nondiffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the UltraBee limiter of [20], [24]. We prove for the UltraBee scheme a property of exact advection for a large s ..."
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Cited by 22 (6 self)
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Abstract We present a nondiffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the UltraBee limiter of [20], [24]. We prove for the UltraBee scheme a property of exact advection for a large set of piecewise constant functions. We prove that the numerical error is uniformly bounded in time for such prepared (i.e. piecewise constant) initial data, and state a conjecture of nondiffusion at infinite time based on some local overcompressivity of the scheme for general initial data. We generalize the scheme to compressible gas dynamics and present some numerical results.
A Critical Analysis of RayleighTaylor Growth Rates
, 2000
"... this paper is to begin a systematic analysis of causes of these discrepancies. To do this, we summarize the results of previous RayleighTaylor instability studies, identify potential sensitive factors in RayleighTaylor simulations, and report on new simulation results designed to quantify the effec ..."
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Cited by 22 (2 self)
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this paper is to begin a systematic analysis of causes of these discrepancies. To do this, we summarize the results of previous RayleighTaylor instability studies, identify potential sensitive factors in RayleighTaylor simulations, and report on new simulation results designed to quantify the effects of a number of these factors.
A Posteriori Error Analysis And Adaptivity For Finite Element Approximations Of Hyperbolic Problems
, 1997
"... this article is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational ..."
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Cited by 16 (4 self)
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this article is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational implementation of the a posteriori error bounds into adaptive finite element algorithms
Optimal Control of Flow With Discontinuities
 Journal of Computational Physics
, 2003
"... Optimal control of the 1D Riemann problem of Euler equations whose solution is characterized by discontinuities is carried out by both nonsmooth and smooth op timization methods. By matching a desired flow to the numerical model for a given time window we effectively change the location of discont ..."
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Cited by 15 (1 self)
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Optimal control of the 1D Riemann problem of Euler equations whose solution is characterized by discontinuities is carried out by both nonsmooth and smooth op timization methods. By matching a desired flow to the numerical model for a given time window we effectively change the location of discontinuities. The control pa rameters are chosen to be the initial values for pressure and density fields. Existence of solutions for the optimal control problem is proven. A high resolution model and a model with artificial viscosity, implementing two different numerical methods, are used to solve the forward model. The cost functional is the weighted difference be tween the numerical values and the observations for pressure, density and velocity. The observations are constructed from the analytical solution. We consider either distributed observations in time or observations calculated at the end of the assimi lation window. We consider two different time horizons and two sets of observations. The gradient (respectively a subgradient) of the cost functional, obtained from the adjoint of the discrete forward model, are employed for the smooth minimization (respectively for the nonsmooth minimization) algorithm. Discontinuity detection improves the performance of the minimizer for the model with artificial viscosity by selecting the points where the shock occurs (and these points are then removed from Preprint submitted to Elsevier Science 26 March 2002 the cost functional and its gradient). The numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well the observations. The algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon.
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.