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Wave Propagation Algorithms for Multidimensional Hyperbolic Systems
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
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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 45 (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 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 25 (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
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
A fourthorder central WENO scheme for multidimensional hyperbolic systems of conservation laws
 SIAM J Sci Comput
"... Abstract. We present the first fourthorder centralscheme for twodimensionalhyperbolic systems of conservation laws. Our new method is based on a central weighted nonoscillatory approach. The heart of our method is the reconstruction step, in which a genuinely twodimensional interpolant is reconst ..."
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Cited by 10 (0 self)
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Abstract. We present the first fourthorder centralscheme for twodimensionalhyperbolic systems of conservation laws. Our new method is based on a central weighted nonoscillatory approach. The heart of our method is the reconstruction step, in which a genuinely twodimensional interpolant is reconstructed from cell averages by taking a convex combination of building blocks in the form of biquadratic polynomials. Similarly to other central schemes, our new method enjoys the simplicity of the blackbox approach. All that is required in order to solve a problem is to supply the flux function and an estimate on the speed of propagation. The highresolution properties of the scheme as well as its resistance to mesh orientation, and the effectiveness of the componentwise approach, are demonstrated in a variety of numericalexamples. Key words. hyperbolic systems, central difference schemes, highorder accuracy, nonoscillatory schemes, weighted essentially nonoscillatory reconstruction, central weighted essentially nonoscillatory reconstruction
Petrova G., A Smoothness Indicator for Adaptive Algorithms for Hyperbolic Systems
 JCP
"... The formation of shock waves in solutions of hyperbolic conservation laws calls for locally adaptive numerical solution algorithms and requires a practical tool for identifying where adaption is needed. In this paper, a new smoothness indicator (SI) is used to identify “rough ” solution regions and ..."
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Cited by 9 (0 self)
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The formation of shock waves in solutions of hyperbolic conservation laws calls for locally adaptive numerical solution algorithms and requires a practical tool for identifying where adaption is needed. In this paper, a new smoothness indicator (SI) is used to identify “rough ” solution regions and is implemented in locally adaptive algorithms. The SI is based on the weak local truncation error of the approximate solution. It was recently reported in S. Karni and A. Kurganov, Local error analysis for approximate solutions of hyperbolic conservation laws, where error analysis and convergence properties were established. The present paper is concerned with its implementation in scheme adaption and mesh adaption algorithms. The SI provides a general framework for adaption and is not restricted to a particular discretization scheme. The implementation in this paper uses the centralupwind scheme of A. Kurganov, S. Noelle, and G. Petrova, SIAM J. Sci. Comput. 23, 707 (2001). The extension of the SI to two space dimensions is given. Numerical results in one and two space dimensions demonstrate the robustness of the proposed SI and its potential in reducing computational costs and improving the resolution of the solution. c ○ 2002 Elsevier Science (USA) Key Words: hyperbolic conservation laws; local truncation error; smoothness indicator; nonoscillatory central schemes. 1.