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52
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 104 (19 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...
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 44 (4 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,
On the construction, comparison, and local characteristic decomposition for highorder central WENO schemes
 J. Comput. Phys
"... In this paper, we review and construct fifth and ninthorder central weighted essentially nonoscillatory (WENO) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge–Kutta methods for solving nonlinear hyperbolic conservation law systems. Negative linear we ..."
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Cited by 27 (1 self)
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In this paper, we review and construct fifth and ninthorder central weighted essentially nonoscillatory (WENO) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge–Kutta methods for solving nonlinear hyperbolic conservation law systems. Negative linear weights appear in such a formulation and they are treated using the technique recently introduced by Shi et al. (J. Comput. Phys. 175, 108 (2002)). We then perform numerical computations and comparisons with the finite difference WENO schemes of Jiang and Shu (J. Comput.
On the reduction of numerical dissipation in centralupwind schemes
 Commun. Comput. Phys
"... We study centralupwind schemes for systems of hyperbolic conservation laws, recently introduced in [A. Kurganov, S. Noelle and G. Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707–740]. Similarly to the staggered central schemes, these schemes are central Godunovtype projectionevolution methods t ..."
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Cited by 24 (6 self)
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We study centralupwind schemes for systems of hyperbolic conservation laws, recently introduced in [A. Kurganov, S. Noelle and G. Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707–740]. Similarly to the staggered central schemes, these schemes are central Godunovtype projectionevolution methods that enjoy the advantages of high resolution, simplicity, universality, and robustness. At the same time, the centralupwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes. In this paper, we present a modification of the onedimensional fully and semidiscrete centralupwind schemes, in which the numerical dissipation is reduced even further. The goal is achieved by a more accurate projection of the evolved quantities onto the original grid. In the semidiscrete case, the reduction of dissipation procedure leads to a new, less dissipative numerical flux. We also extend the new semidiscrete scheme to the twodimensional case via the rigorous, genuinely multidimensional derivation. The new semidiscrete schemes are tested on a number of numerical examples, where one can observe an improved resolution, especially of the contact waves. 1
A Variational Technique for Time Consistent Tracking of Curves and Motion
 J MATH IMAGING VIS
"... In this paper, a new framework for the tracking of closed curves and their associated motion fields is described. The proposed method enables a continuous tracking along an image sequence of both a deformable curve and its velocity field. Such an approach is formalized through the minimization of a ..."
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Cited by 18 (7 self)
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In this paper, a new framework for the tracking of closed curves and their associated motion fields is described. The proposed method enables a continuous tracking along an image sequence of both a deformable curve and its velocity field. Such an approach is formalized through the minimization of a global spatiotemporal continuous cost functional, w.r.t a set of variables representing the curve and its related motion field. The resulting minimization process relies on optimal control approach and consists in a forward integration of an evolution law followed by a backward integration of an adjoint evolution model. This latter pde includes a term related to the discrepancy between the current estimation of the state variable and discrete noisy measurements of the system. The closed curves are represented through implicit surface modeling, whereas the motion is described either by a vector field or through vorticity and divergence maps depending on the kind of targeted applications. The efficiency of the approach is demonstrated on two types of image sequences showing deformable objects and fluid motions.
Central schemes on overlapping cells
, 2005
"... Nessyahu and Tadmor's central scheme [J. Comput. Phys. 87 (1990)] has the benefit of not using Riemann solvers for solving hyperbolic conservation laws. But the staggered averaging causes large dissipation when the time step size is small compared to the mesh size. The recent work of Kurganov ..."
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Cited by 17 (5 self)
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Nessyahu and Tadmor's central scheme [J. Comput. Phys. 87 (1990)] has the benefit of not using Riemann solvers for solving hyperbolic conservation laws. But the staggered averaging causes large dissipation when the time step size is small compared to the mesh size. The recent work of Kurganov and Tadmor [J. Comput. Phys. 160 (2000)] overcomes this problem by using a variable control volume and results in semidiscrete and fully discrete nonstaggered schemes. Motivated by this work, we introduce overlapping cell averages of the solution at the same discrete time level, and develop a simple alternative technique to control the O(1/Dt) dependence of the dissipation. The semidiscrete form of the central scheme can also be obtained to which the TVD Runge–Kutta time discretization methods of Shu and Osher [J. Comput. Phys. 77 (1988)] or other stable and sufficiently accurate ODE solvers can be applied. This technique is essentially independent of the reconstruction and the shape of the mesh. The overlapping cell representation of the solution also opens new possibilities for reconstructions. Generally speaking, more compact reconstruction can be achieved. In the following, schemes of up to fifth order in 1D and third order in 2D have been developed. We demonstrate through numerical examples that by combining two classes of the overlapping cells in the reconstruction we can achieve higher resolution.
Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction
 SIAM J. Numer. Anal
, 2007
"... Abstract. The central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990)] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of ..."
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Cited by 16 (8 self)
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Abstract. The central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990)] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys, 160 (2000)] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu; J. Comput. Phys, 209 (2005)]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunovtype finite volume schemes. Overlapping cells lend themselves to the development of a centraltype discontinuous Galerkin (DG) method, following the series of work by Cockburn and Shu [J. Comput. Phys. 141 (1998)] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities, beyond those of Godunovtype schemes. In particular, the
A Parallel MeshAdaptive Framework for Hyperbolic Conservation Laws, ArXiv Physics eprints
, 2006
"... We report on the development of a computational framework for the parallel, meshadaptive solution of systems of hyperbolic conservation laws like the timedependent Euler equations in compressible gas dynamics or MagnetoHydrodynamics (MHD) and similar models in plasma physics. Local mesh refinemen ..."
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Cited by 11 (3 self)
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We report on the development of a computational framework for the parallel, meshadaptive solution of systems of hyperbolic conservation laws like the timedependent Euler equations in compressible gas dynamics or MagnetoHydrodynamics (MHD) and similar models in plasma physics. Local mesh refinement is realized by the recursive bisection of grid blocks along each spatial dimension, implemented numerical schemes include standard finitedifferences as well as shockcapturing central schemes, both in connection with RungeKutta type integrators. Parallel execution is achieved through a configurable hybrid of POSIXmultithreading and MPIdistribution with dynamic load balancing. One two and threedimensional test computations for the Euler equations have been carried out and show good parallel scaling behavior. The Racoon framework is currently used to study the formation of singularities in plasmas and fluids.
Centralupwind schemes on triangular grids for hyperbolic systems of conservation laws
 Numer. Methods Partial Differential Eq
"... We present a family of centralupwind schemes on general triangular grids for solving twodimensional systems of conservation laws. The new schemes enjoy the main advantages of the Godunovtype central schemes—simplicity, universality, and robustness and can be applied to problems with complicated g ..."
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Cited by 11 (1 self)
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We present a family of centralupwind schemes on general triangular grids for solving twodimensional systems of conservation laws. The new schemes enjoy the main advantages of the Godunovtype central schemes—simplicity, universality, and robustness and can be applied to problems with complicated geometries. The “triangular ” centralupwind schemes are based on the use of the directional local speeds of propagation and are a generalization of the centralupwind schemes on rectangular grids, recently introduced in Kurganov et al. [SIAM J Sci Comput 23 (2001), 707–740]. We test a secondorder version of the proposed scheme on various examples. The main purpose of the numerical experiments is to demonstrate the potential of our method. The more universal “triangular ” centralupwind schemes provide the same high accuracy and resolution as the original, “rectangular ” ones, and at the same time, they can be used to solve hyperbolic systems of conservation laws on complicated domains, where the implementation of triangular or mixed grids is advantageous. © 2004 Wiley Periodicals, Inc. Numer Methods Partial