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Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 270 (26 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi 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 self-contained. 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 high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector 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 133 (11 self)
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We construct, analyze, and implement a new nonoscillatory high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewise-linear 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 second-order, nonoscillatory central scheme, a natural extension of the one-dimensional second-order central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408--448]. As in the one-dimensional case, the main feature of our two-dimensional scheme is simplicity. In particular, this central scheme does not require the intricate and time-consuming (approximate) Riemann solvers which are essential for the high-resolution upwind schemes; in fact, even the com...
Semi-Discrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations
- SIAM J. Sci. Comput
, 2000
"... We introduce new Godunov-type semi-discrete central schemes for hyperbolic systems of conservation laws and Hamilton-Jacobi 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 116 (22 self)
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We introduce new Godunov-type semi-discrete central schemes for hyperbolic systems of conservation laws and Hamilton-Jacobi 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 one-sided local speeds. This is the reason why we call them central-upwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton-Jacobi 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 two-phase 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 han-dling of topological changes ..."
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Cited by 73 (15 self)
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Since the seminal work of [92] on coupling the level set method of [69] to the equations for two-phase 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 han-dling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interest-ingly, 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 in-compressible flow due to both its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical meth-ods such as Hamilton-Jacobi 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].
High-Resolution 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 60 (15 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. 408--463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 1892--1917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemann-solver-free 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.
Compact central WENO schemes for multidimensional conservation laws
- SIAM J. Sci. Comput
, 2000
"... We present new third- and fifth-order Godunov-type central schemes for approximating solutions of the Hamilton-Jacobi (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 60 (12 self)
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We present new third- and fifth-order Godunov-type central schemes for approximating solutions of the Hamilton-Jacobi (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 third-order scheme: one scheme that is based on a genuinely two-dimensional Central WENO reconstruction, and another scheme that is based on a simpler dimension-by-dimension reconstruction. The simpler dimension-by-dimension variant is then extended to a multi-dimensional fifth-order scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes. Key words. Hamilton-Jacobi equations, central schemes, high order, WENO, CWENO.
Central WENO Schemes for Hyperbolic Systems of Conservation Laws
- MATH. MODEL. NUMER. ANAL
, 2001
"... We present a family of high-order, essentially non-oscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially NonOscillatory (WENO) reconstruction of point-values from cell-averages ..."
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Cited by 59 (13 self)
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We present a family of high-order, essentially non-oscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially NonOscillatory (WENO) reconstruction of point-values from cell-averages, which is then followed by an accurate approximation of the fluxes via a natural continuous extension of Runge-Kutta solvers. We explicitly construct the third and fourthorder scheme and demonstrate their high-resolution properties in several numerical tests.
An Eulerian formulation for solving partial differential equations along a moving interface
- Jardin Botanique - BP 101 - 54602 Villers-ls-Nancy Cedex (France) Unit de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unit de recherche INRIA Rhne-Alpes : 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 Navier-Stokes NS and Cahn-Hilliard CH system that has an associated discrete energy functional. This system provides a diffuse-interface description of binary fluid flows with compressible or inco ..."
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Cited by 45 (7 self)
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We develop a conservative, second order accurate fully implicit discretization in two dimensions of the Navier-Stokes NS and Cahn-Hilliard CH system that has an associated discrete energy functional. This system provides a diffuse-interface 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 density-matched components. The scheme, however, has a straightforward extension to multi-component systems. To efficiently solve the discrete system at the implicit time-level, 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
