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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...
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,
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 13 (6 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 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 8 (7 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
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 and ..."
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Cited by 6 (3 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.
R.: Racoon: A parallel meshadaptive framework for hyperbolic conservation laws. Parallel Computing 31
, 2005
"... 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 ..."
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Cited by 6 (2 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. Key words: AMR, mesh refinement, hybrid parallelization, multithread, MPI, load balancing
Nonoscillatory hierarchical reconstruction for central and finite volume schemes
 Comm. Comput. Phys
"... Abstract. This is the continuation of the paper ”Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction ” by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite v ..."
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Cited by 5 (5 self)
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Abstract. This is the continuation of the paper ”Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction ” by the same authors. The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite volume schemes on nonstaggered grids. This takes a new finite volume approach for approximating nonsmooth solutions. A critical step for highorder finite volume schemes is to reconstruct a nonoscillatory high degree polynomial approximation in each cell out of nearby cell averages. In the paper this procedure is accomplished in two steps: first to reconstruct a high degree polynomial in each cell by using e.g., a central reconstruction, which is easy to do despite the fact that the reconstructed polynomial could be oscillatory; then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution. All numerical computations for systems of conservation laws are performed without characteristic decomposition. In particular, we demonstrate that this new approach can generate essentially nonoscillatory solutions even for 5thorder schemes without characteristic decomposition.
Staggered Finite Difference Schemes for Conservation Laws
 Brown University
"... Key words. Conservation laws, balance laws, finite difference schemes, highorder accuracy, central schemes. Here we show how to construct finitedifference shockcapturing central schemes on staggered grids. Staggered schemes may have better resolution of the corresponding unstaggered schemes of th ..."
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Cited by 2 (1 self)
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Key words. Conservation laws, balance laws, finite difference schemes, highorder accuracy, central schemes. Here we show how to construct finitedifference shockcapturing central schemes on staggered grids. Staggered schemes may have better resolution of the corresponding unstaggered schemes of the same order. They are based on high order non oscillatory reconstruction (ENO or WENO), and a suitable ODE solver for the computation of the integral of the flux. Although they suffer a more severe stability restriction, they do not require a numerical flux function. A comparison between central finite volume and finite difference, on staggered and non staggered grids, is reported. 1
A Central Scheme for Shallow Water Flows along Channels with Irregular Geometry
, 2007
"... We present a new semidiscrete central scheme for onedimensional shallow water flows along channels with nonuniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steadystates of rest (i.e. it is wellbalanced). Along ..."
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Cited by 1 (0 self)
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We present a new semidiscrete central scheme for onedimensional shallow water flows along channels with nonuniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steadystates of rest (i.e. it is wellbalanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm. AMS subject classification: Primary 65M99; Secondary 35L65 Key words: Hyperbolic systems of conservation and balance laws, semidiscrete schemes, SaintVenant system of Shallow Water equations, nonoscillatory reconstructions, channels with irregular geometry. 1 The Shallowwater Model We consider the shallow water equations along channels with nonuniform rectangular cross sections and bottom topography. The model describes flows that are nearly horizontal and can be obtained by averaging the Euler equations over the channel cross section [3], resulting in the balance law ∂A ∂t