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412
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 275 (28 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically selfcontained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
Nonoscillatory Central Schemes For Multidimensional Hyperbolic Conservation Laws
 SIAM J. Sci. Comput
, 1998
"... We construct, analyze, and implement a new nonoscillatory highresolution scheme for twodimensional hyperbolic conservation laws. The scheme is a predictorcorrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory ..."
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Cited by 138 (15 self)
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We construct, analyze, and implement a new nonoscillatory highresolution scheme for twodimensional hyperbolic conservation laws. The scheme is a predictorcorrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewiselinear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a secondorder, nonoscillatory central scheme, a natural extension of the onedimensional secondorder central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408448]. As in the onedimensional case, the main feature of our twodimensional scheme is simplicity. In particular, this central scheme does not require the intricate and timeconsuming (approximate) Riemann solvers which are essential for the highresolution upwind schemes; in fact, even the com...
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 115 (20 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...
Spatially adaptive techniques for level set methods and incompressible flow
 Comput. Fluids
"... Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes ..."
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Cited by 74 (15 self)
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Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [92] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as HamiltonJacobi WENO [46], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [27] and the coupled level set volume of fluid method [91], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [56].
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 63 (14 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.
HighResolution Nonoscillatory Central Schemes With Nonstaggered Grids For Hyperbolic Conservation Laws
 SIAM J. Numer. Anal
, 1998
"... We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recent ..."
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Cited by 61 (17 self)
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We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 18921917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemannsolverfree recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.
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 59 (4 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.
An Eulerian formulation for solving partial differential equations along a moving interface
 Jardin Botanique  BP 101  54602 VillerslsNancy Cedex (France) Unit de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu  35042 Rennes Cedex (France) Unit de recherche INRIA RhneAlpes : 655, avenue de l'Europe  38334 Montbonnot Saint
, 2002
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Conservative multigrid methods for Cahn–Hilliard fluids
 J. Comput. Phys
"... We develop a conservative, second order accurate fully implicit discretization in two dimensions of the NavierStokes NS and CahnHilliard CH system that has an associated discrete energy functional. This system provides a diffuseinterface description of binary fluid flows with compressible or inco ..."
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Cited by 47 (7 self)
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We develop a conservative, second order accurate fully implicit discretization in two dimensions of the NavierStokes NS and CahnHilliard CH system that has an associated discrete energy functional. This system provides a diffuseinterface description of binary fluid flows with compressible or incompressible flow components [44,4]. In this work, we focus on the case of flows containing two immiscible, incompressible and densitymatched components. The scheme, however, has a straightforward extension to multicomponent systems. To efficiently solve the discrete system at the implicit timelevel, we develop a nonlinear multigrid method to solve the CH equation which is then coupled to a projection method that is used to solve the NS equation. We analyze and prove convergence of the scheme in the absence of flow. We demonstrate convergence of our scheme numerically in both the presence and absence of flow and perform simulations of phase separation via spinodal decomposition. We examine the separate effects of surface tension and external flow on the decomposition. We find surface tension driven flow alone increases coalescence rates through the retraction of interfaces. When there is an external shear flow, the evolution