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## Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws (1998)

Citations: | 270 - 26 self |

### Citations

1481 |
Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
- Osher, Sethian
- 1988
(Show Context)
Citation Context ...d in [35], [83], [70] and [43]. Multidimensional ENO schemes based on general triangulation were developed in [1]. ENO and WENO schemes for Hamilton-Jacobi type equations were designed and applied in =-=[59]-=-, [60], [50] and [45]. ENO schemes using one-sided Jocobians for field by field decomposition, which improves the robustness for calculations of systems, were discussed in [25]. Combination of ENO wit... |

1010 | Approximate riemann solvers, parameter vectors, and difference schemes
- ROE
- 1981
(Show Context)
Citation Context ...v - i+ 1 2 for the numerical fluxsf i+ 1 2 ; 22 . if a i+ 1 2s0, then the wind blows from the right to the left. We would use v + i+ 1 2 for the numerical fluxsf i+ 1 2 . This produces the Roe scheme =-=[62] at the fi-=-rst order level. For this reason, the ENO scheme based on this approach was termed "ENO-Roe" in [70]. In summary, to build a finite di#erence ENO scheme (2.73) using the ENO-Roe approach,giv... |

909 |
Efficient implementation of essentially non-oscillatory shock capturing scheme,
- Shu, Osher
- 1988
(Show Context)
Citation Context ...he area of its applications. ENO schemes based on point values and TVD Runge-Kutta time discretizations, which can save computational costs significantly for multi space dimensions, were developed in =-=[69]-=- and [70]. Later biasing in the stencil choosing process to enhance stability and accuracy were developed in [28] and [67]. Weighted ENO (WENO) schemes were developed, using a convex combination of al... |

737 |
Hyperbolic systems of conservation laws
- Lax
- 1957
(Show Context)
Citation Context ...he two fourth order ENO schemes behave similarly. We thus use the simple and inexpensive Lax-Friedrichs flux in most of our high order calculations. We remark that, by the classic Lax-Wendro# theorem =-=[51]-=-, the solution to the conservative scheme (2.68), if converges, will converge to a weak solution of (2.65). In summary, to build a finite volume ENO scheme (2.68), given the cell averages {u i } (we w... |

607 |
A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow,
- Sussman, Smereka, et al.
- 1994
(Show Context)
Citation Context ...12#x = 3 32 ; Bottom left: ENO with 512 2 points, # function width # = 24#x = 3 32 ; Bottom right: ENO with 1024 2 points, # function width # = 48#x = 3 32 . The # function is approximated as in [61],=-=[77]-=- by # # (#)= # 1 2# (1 + cos # ## # # )if|#|s0otherwise (5.32) Fo r fi xed #, there is convergence as #x # 0 to a smooth solution. One can then take # # 0. This two step limit is very costly to implem... |

555 |
Visconsity solutions of Hamilton-Jacobi equations
- Crandall, Lions
- 1983
(Show Context)
Citation Context ...erentiable, (4.34) is satisfied in the classical sense. Viscosity solution defined this way exists and is unique. For details and equivalent definitions of viscosity solutions, see Crandall and Lions =-=[22]-=-. Hamilton-Jacobi equations are actually easier to solve than conservation laws, because the solutions are typically continuous (only the derivatives are discontinuous). As before, given mesh sizes #x... |

508 | Runge-Kutta discontinuous Galerkin methods for convectiondominated problems,
- Cockburn, Shu
- 2001
(Show Context)
Citation Context ...d in this section can also be applied to other types of spatial discretizations using the method of lines approach, such as various TVD and TVB schemes [52, 78, 65] and discontinuous Galerkin methods =-=[18, 19, 20, 21, 17]-=-. 4.2.1. TVD Runge-Kutta methods. A class of TVD (total variation diminishing) high order RungeKutta methods is developed in [69] and further in [29]. These Runge-Kutta methods are used to solve a sys... |

489 |
High Resolution Schemes for Hyperbolic Conservation Laws.
- Harten
- 1983
(Show Context)
Citation Context ... i+1 - u i ) The monotone flux h is the Godunov flux defined by (2.70), and the minmod function is given by minmod(a, b)= sign(a)+sign(b) 2 min(|a|, |b|). It is easy to prove, by using Harten's Lemma =-=[33]-=-, that the Euler forward time discretization with this second order MUSCL spatial operator is TVD under the CFL condition (4.6): #t # #x 2max j |u n j | (4.21) Thus #t = #x 2max j |u n j | will be use... |

432 |
TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework,
- Cockburn, Shu
- 1989
(Show Context)
Citation Context ...d in this section can also be applied to other types of spatial discretizations using the method of lines approach, such as various TVD and TVB schemes [52, 78, 65] and discontinuous Galerkin methods =-=[18, 19, 20, 21, 17]-=-. 4.2.1. TVD Runge-Kutta methods. A class of TVD (total variation diminishing) high order RungeKutta methods is developed in [69] and further in [29]. These Runge-Kutta methods are used to solve a sys... |

412 | Efficient implementation of weighted ENO schemes. - Jiang, CW - 1996 |

334 |
Numerical analysis of blood flow in heart
- Peskin
- 1977
(Show Context)
Citation Context ... # = 12#x = 3 32 ; Bottom left: ENO with 512 2 points, # function width # = 24#x = 3 32 ; Bottom right: ENO with 1024 2 points, # function width # = 48#x = 3 32 . The # function is approximated as in =-=[61]-=-,[77] by # # (#)= # 1 2# (1 + cos # ## # # )if|#|s0otherwise (5.32) Fo r fi xed #, there is convergence as #x # 0 to a smooth solution. One can then take # # 0. This two step limit is very costly to i... |

326 | Weighted essentially nonoscillatory schemes.
- Liu, Osher, et al.
- 1994
(Show Context)
Citation Context ...ce stability and accuracy were developed in [28] and [67]. Weighted ENO (WENO) schemes were developed, using a convex combination of all candidate stencils instead of just one as in the original ENO, =-=[53]-=-, [43]. ENO schemes based on other than polynomial building blocks were constructed in [40], [16]. Sub-cell resolution and artificial compression to sharpen contact discontinuities were studied in [35... |

309 |
High resolution schemes using flux limiters for hyperbolic conservation laws”,
- Sweby
- 1984
(Show Context)
Citation Context ... of time discretization. The techniques discussed in this section can also be applied to other types of spatial discretizations using the method of lines approach, such as various TVD and TVB schemes =-=[52, 78, 65]-=- and discontinuous Galerkin methods [18, 19, 20, 21, 17]. 4.2.1. TVD Runge-Kutta methods. A class of TVD (total variation diminishing) high order RungeKutta methods is developed in [69] and further in... |

252 |
The numerical simulations of two-dimensional fluid flows with strong shock.
- Woodward, Collela
- 1984
(Show Context)
Citation Context ...er WENO with 800 points; middle: fifth order WENO with 1200 points; bottom: second order TVD with 2000 points. 4. Forward facing step problem. This is a standard test case for high resolution schemes =-=[82]-=-. However, second order methods usually already work well. High order methods might have some advantage in resolving the slip lines. The set up of the problem is the following: the wind tunnel is 1 le... |

249 |
Level Set Methods: Evolving Interfaces
- Sethian
- 1996
(Show Context)
Citation Context ...ems [12], [27], [43]; to incompressible flow problems [26], [31]; to viscoelasticity equations with fading memory [72]; to semi-conductor device simulation [28], [41], [42]; to image processing [59], =-=[64]-=-, [73]; etc. This list is definitely incomplete and may be biased by the author's own research experience, but one can already see that ENO and WENO have been applied quite extensively in many di#eren... |

229 | Weighted ENO Schemes for HamiltonJacobi Equations,”
- Jiang, Peng
- 2000
(Show Context)
Citation Context ... and [43]. Multidimensional ENO schemes based on general triangulation were developed in [1]. ENO and WENO schemes for Hamilton-Jacobi type equations were designed and applied in [59], [60], [50] and =-=[45]-=-. ENO schemes using one-sided Jocobians for field by field decomposition, which improves the robustness for calculations of systems, were discussed in [25]. Combination of ENO with multiresolution ide... |

215 | High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, - Osher, Shu - 1991 |

213 |
Two approximations of solutions of Hamilton-Jacobi equations,
- Crandall, Lions
- 1984
(Show Context)
Citation Context ...+1 ij = G(# n i-p,j-r , ,# n i+q,j+s ) (4.37) where G is a non-decreasing function of each argument, is called a first order scheme, although the provable order of accuracy in the L# norm is just 1 2 =-=[23]-=-. In the semi-discrete formulation, a five point monotone scheme (it does not pay to use more points for a monotone scheme because the order of accuracy of a monotone scheme is at most one [36]) is of... |

171 | Total variation diminishing Runge-Kutta schemes,
- Gottlieb, Shu
- 1998
(Show Context)
Citation Context ...and discontinuous Galerkin methods [18, 19, 20, 21, 17]. 4.2.1. TVD Runge-Kutta methods. A class of TVD (total variation diminishing) high order RungeKutta methods is developed in [69] and further in =-=[29]-=-. These Runge-Kutta methods are used to solve a system of initial value problems of ODEs written as: u t = L(u), (4.2) resulting from a method of lines spatial approximation to a PDE such as: u t = -f... |

155 |
Total-Variation-Diminishing Time Discretizations,”
- Shu
- 1988
(Show Context)
Citation Context ...n in many cases, such as various TVD and TVB schemes in 1D (where the norm is the total variation norm) and in multi dimensions (where the norm is the L # norm), see, e.g. [52, 78, 65]. Originally in =-=[69, 66] the norm -=-in (4.5) was chosen to be the total variation norm, hence the terminology "TVD time discretization". As it stands, the TVD high order time discretization defined above maintains stability in... |

139 |
Numerical methods for Conservations Laws. Birkhauser,
- LeVeque
- 1992
(Show Context)
Citation Context ...s to first order near smooth extrema. We will not discuss the method of adding explicit artificial viscosity or the TVD method in these lecture notes. We refer to the books by Sod [75] and by LeVeque =-=[52]-=-, and the references listed therein, for details. The ENO idea proposed in [38] seems to be the first successful attempt to obtain a self similar (i.e. no mesh size dependent parameter), uniformly hig... |

128 |
Uniformly High Order Essentially Non-Oscillatory Schemes, III,
- Harten, Enquist, et al.
- 1987
(Show Context)
Citation Context ...Subject classification. Applied and Numerical Mathematics 1. Introduction. ENO (Essentially Non-Oscillatory) schemes started with the classic paper of Harten, Engquist, Osher and Chakravarthy in 1987 =-=[38]-=-. This paper has been cited at least 144 times by early 1997, according to the ISI database. The Journal of Computational Physics decided to republish this classic paper as part of the celebration of ... |

111 |
Uniformly high-order accurate nonoscillatory schemes,
- Harten, Osher
- 1987
(Show Context)
Citation Context ...neric solution for hyperbolic conservation laws is in the class of piecewise smooth functions. The reconstruction in [38] is a natural extension of an earlier second order version of Harten and Osher =-=[37]-=-. In [38], Harten, Engquist, Osher and Chakravarthy investigated di#erent ways of measuring local smoothness to determine the local stencil, and developed a hierarchy that begins with one or two cells... |

94 |
Towards the ultimate conservative di erence scheme. IV. A new approach to numerical convection.
- Leer
- 1977
(Show Context)
Citation Context ...du # 0. (2.76) We would need the positive and negative fluxes fs(u) to have as many derivatives as the order of the scheme. This unfortunately rules out many popular fluxes (such as those of van Leer =-=[79]-=- and Osher [58]) for high order methods in this framework. 23 The simplest smooth splitting is the Lax-Friedrichs splitting: fs(u)= 1 2 (f(u) #u) (2.77) where # is again taken as # =max u |f # (u)| ov... |

94 |
Low-storage Runge-Kutta schemes”
- Williamson
- 1980
(Show Context)
Citation Context ...also given in [69]. For large scale scientific computing in three space dimensions, storage is usually a paramount consideration. There are therefore discussions about low storage Runge-Kutta methods =-=[81]-=-, [10], which only require 2 storage units per ODE equation. In [29], we considered the TVD properties among such low storage Runge-Kutta methods and found third order low storage TVD Runge-Kutta meth... |

93 |
A study of singularity formation in a vortex sheet by the point-vortex approximation
- Krasny
- 1986
(Show Context)
Citation Context ...ensions, i.e. P (#)=#(#) in (5.31). The three dimensional case is defined in detail later. The evolution of the vortex sheet in the Lagrangian framework has been considered by various authors. Krasny =-=[47]-=-, [48] has computed vortex sheet roll-up using vortex blobs and point vortices with filtering. Baker and Shelley [4] have approximated the vortex sheet by a layer of constant vorticity which they comp... |

76 | Stability theory of difference approximations for mixed initial boundary value problems. - Gustafsson, Kreiss, et al. - 1972 |

74 |
On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation
- Abgrall
- 1994
(Show Context)
Citation Context ...-cell resolution and artificial compression to sharpen contact discontinuities were studied in [35], [83], [70] and [43]. Multidimensional ENO schemes based on general triangulation were developed in =-=[1]-=-. ENO and WENO schemes for Hamilton-Jacobi type equations were designed and applied in [59], [60], [50] and [45]. ENO schemes using one-sided Jocobians for field by field decomposition, which improves... |

74 |
Desingularization of Periodic Vortex Sheet Roll-up.
- Krasny
- 1986
(Show Context)
Citation Context ...s, i.e. P (#)=#(#) in (5.31). The three dimensional case is defined in detail later. The evolution of the vortex sheet in the Lagrangian framework has been considered by various authors. Krasny [47], =-=[48]-=- has computed vortex sheet roll-up using vortex blobs and point vortices with filtering. Baker and Shelley [4] have approximated the vortex sheet by a layer of constant vorticity which they computed b... |

70 |
schemes with subcell resolution
- ENO
- 1989
(Show Context)
Citation Context ...53], [43]. ENO schemes based on other than polynomial building blocks were constructed in [40], [16]. Sub-cell resolution and artificial compression to sharpen contact discontinuities were studied in =-=[35]-=-, [83], [70] and [43]. Multidimensional ENO schemes based on general triangulation were developed in [1]. ENO and WENO schemes for Hamilton-Jacobi type equations were designed and applied in [59], [60... |

67 |
Fourth-order 2N-storage Runge-Kutta schemes.
- Carpenter, Kennedy
- 1994
(Show Context)
Citation Context ...iven in [69]. For large scale scientific computing in three space dimensions, storage is usually a paramount consideration. There are therefore discussions about low storage Runge-Kutta methods [81], =-=[10]-=-, which only require 2 storage units per ODE equation. In [29], we considered the TVD properties among such low storage Runge-Kutta methods and found third order low storage TVD Runge-Kutta methods. 4... |

55 |
A second order projection method for the incompressible Navier-Stokes equations.
- Bell, Colella, et al.
- 1989
(Show Context)
Citation Context ...thout fully resolving the flow, but still get back some useful information. A pioneer work in applying shock capturing compressible flow techniques to incompressible flow is by Bell, Colella and Glaz =-=[6]-=-, in which they considered a second order Godunov type discretization, investigated the projection into divergence-free velocity fields for general boundary conditions, and discussed accuracy of time ... |

53 |
A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems,
- Adams, Shariff
- 1996
(Show Context)
Citation Context ...19 grid points. Top: fifth order WENO; bottom: fourth order ENO. 58 The model problem we use describes the interaction between a stationary shock and a vortex. The computational domain is taken to be =-=[0, 2]-=- [0, 1]. A stationary Mach 1.1 shock is positioned at x =0.5 and normal to the x-axis. Its left state is (#, u, v, P)=(1, # #,0, 1). A small vortex is superposed to the flow left to the shock and cent... |

50 |
Shock waves and reaction-diusion equations
- Smoller
- 1994
(Show Context)
Citation Context ..., which is used very often in practice. However, it is usually very costly to get this solution (for Euler equations of compressible gas, an iterative procedure is needed to obtain this solution, see =-=[74]-=-). In practice, approximate Riemann solvers are usually good enough. As in the scalar case, the quality of the solution is usually very sensitive to the choice of approximate Riemann solvers for lower... |

48 |
The Artificial Compression Method for Computation of Shocks and Contact Discontinuities: III. Self-Adjusting Hybrid Schemes
- Harten
- 1978
(Show Context)
Citation Context ...ing problems (we termed it geometric ENO, or GENO). 4.1.2. Artificial compression. Another very useful idea to sharpen a contact discontinuity is the artificial compression, first developed by Harten =-=[32]-=- and further improved by Yang [83]. The idea is to increase the magnitude of the slope of a reconstruction, of course subject to certain monotonicity restrictions, near such a discontinuity. Notice th... |

48 |
The Fourier method for nonsmooth initial data
- MCDONOUGH, J, et al.
- 1978
(Show Context)
Citation Context ...r central di erences which, when used twice, will produce the fourth order central di erence approximation 16(wi+1+wi;1);(wi+2+wi;2);30wi 12 x2 for wxx. High order lters, such as the exponential lter =-=[55]-=-, [46]: k x ; ( k = e Nx )2p � 74 y ; ( l Ny l = e )2p (5:14)swhere 2p is the order of the lter and is chosen so that e ; is machine zero, can be used to enhance the stability while keeping at least 2... |

45 | Numerical experiments on the accuracy of ENO and modified ENO schemes - Shu - 1990 |

44 |
Ecient implementation of weighted ENO schemes,
- Shu, Jiang
- 1996
(Show Context)
Citation Context ...bility and accuracy were developed in [28] and [67]. Weighted ENO (WENO) schemes were developed, using a convex combination of all candidate stencils instead of just one as in the original ENO, [53], =-=[43]-=-. ENO schemes based on other than polynomial building blocks were constructed in [40], [16]. Sub-cell resolution and artificial compression to sharpen contact discontinuities were studied in [35], [83... |

40 | Geometric shock-capturing ENO schemes for subpixel interpolation, computation and curve evolution,” Graph.
- Siddiqi, Kimia, et al.
- 1997
(Show Context)
Citation Context ...2], [27], [43]; to incompressible flow problems [26], [31]; to viscoelasticity equations with fading memory [72]; to semi-conductor device simulation [28], [41], [42]; to image processing [59], [64], =-=[73]-=-; etc. This list is definitely incomplete and may be biased by the author's own research experience, but one can already see that ENO and WENO have been applied quite extensively in many di#erent fiel... |

38 |
Upwind Schemes and Boundary Conditions with Applications to Euler equations in
- Osher, Chakravarthy
- 1983
(Show Context)
Citation Context ...We would need the positive and negative fluxes fs(u) to have as many derivatives as the order of the scheme. This unfortunately rules out many popular fluxes (such as those of van Leer [79] and Osher =-=[58]-=-) for high order methods in this framework. 23 The simplest smooth splitting is the Lax-Friedrichs splitting: fs(u)= 1 2 (f(u) #u) (2.77) where # is again taken as # =max u |f # (u)| over the relevant... |

38 | On finite difference approximations and entropy conditions - Harten, Hyman, et al. - 1976 |

37 | Capturing shock reflections: an improved flux formula
- DONAT, MARQUINA
- 1996
(Show Context)
Citation Context ...designed and applied in [59], [60], [50] and [45]. ENO schemes using one-sided Jocobians for field by field decomposition, which improves the robustness for calculations of systems, were discussed in =-=[25]-=-. Combination of ENO with multiresolution ideas was pursued in [7]. Combination of ENO with spectral method using a domain decomposition approach was carried out in [8]. On the application side, ENO a... |

35 |
Solution of the hydrodynamic device model using high-order nonoscillatory shock capturing algorithms
- Fatemi, Jerome, et al.
- 1991
(Show Context)
Citation Context ...n save computational costs significantly for multi space dimensions, were developed in [69] and [70]. Later biasing in the stencil choosing process to enhance stability and accuracy were developed in =-=[28]-=- and [67]. Weighted ENO (WENO) schemes were developed, using a convex combination of all candidate stencils instead of just one as in the original ENO, [53], [43]. ENO schemes based on other than poly... |

35 |
Numerical Methods in Fluid Dynamics,
- Sod
- 1985
(Show Context)
Citation Context ...cessarily degenerates to first order near smooth extrema. We will not discuss the method of adding explicit artificial viscosity or the TVD method in these lecture notes. We refer to the books by Sod =-=[75]-=- and by LeVeque [52], and the references listed therein, for details. The ENO idea proposed in [38] seems to be the first successful attempt to obtain a self similar (i.e. no mesh size dependent param... |

34 |
Nonoscillatory Central Dierencing for Hyperbolic Conservation
- Nessyahu, Tadmor
- 1990
(Show Context)
Citation Context ...Shu [43] used smooth indicators based on density and pressure to perform the so-called pseudo characteristic decompositions. There are also second and sometimes third order component ENO type schemes =-=[56]-=-, [54], with limited success for higher order methods. 32 3. Multi Space Dimensions. 3.1. Reconstruction and Approximation in Multi Dimensions. In this section we describe how the ideas of reconstruct... |

32 | Interaction of a shock with a longitudinal vortex
- Erlebacher, Hussaini, et al.
- 1997
(Show Context)
Citation Context ...0], [71], [2]; to the direct 2 simulation of compressible turbulence [71], [80], [49]; to relativistic hydrodynamics equations [24]; to shock vortex interactions and other gas dynamics problems [12], =-=[27]-=-, [43]; to incompressible flow problems [26], [31]; to viscoelasticity equations with fading memory [72]; to semi-conductor device simulation [28], [41], [42]; to image processing [59], [64], [73]; et... |

32 |
Energy models for one-carrier transport in semiconductor devices
- Jerome, Shu
(Show Context)
Citation Context ...eractions and other gas dynamics problems [12], [27], [43]; to incompressible flow problems [26], [31]; to viscoelasticity equations with fading memory [72]; to semi-conductor device simulation [28], =-=[41]-=-, [42]; to image processing [59], [64], [73]; etc. This list is definitely incomplete and may be biased by the author's own research experience, but one can already see that ENO and WENO have been app... |

31 | Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids.
- Liu, Osher
- 1998
(Show Context)
Citation Context ...3] used smooth indicators based on density and pressure to perform the so-called pseudo characteristic decompositions. There are also second and sometimes third order component ENO type schemes [56], =-=[54]-=-, with limited success for higher order methods. 32 3. Multi Space Dimensions. 3.1. Reconstruction and Approximation in Multi Dimensions. In this section we describe how the ideas of reconstruction an... |

29 |
The nonconvex multi-dimensional Riemann problem for Hamilton–Jacobi equations
- Bardi, Osher
- 1991
(Show Context)
Citation Context .... Whenever applicable it should be used. This flux is similar to the Engquist-Osher monotone flux (2.71) for the conservation laws. 2. For the general H we can always use the Godunov type Hamiltonian =-=[5]-=-, [60]: H G (u + ,u - ,v + ,v - )=ext u#I(u - ,u + ) ext v#I(v - ,v + ) H(u, v) (4.42) where the extrema are defined by ext u#I(a,b) = # min a#u#b if a # b max b#u#a if a>b (4.43) Godunov Hamiltonian ... |

25 |
On the connection between thin vortex layers and vortex sheets
- Baker, Shelley
- 1990
(Show Context)
Citation Context ...second order methods usually again already work well. High order methods might have some advantage in resolving the flow below the Mach stem. The computational domain for this problem is chosen to be =-=[0, 4]-=- [0, 1], although only part of it, [0, 3] [0, 1], is shown [82]. The reflecting wall lies at the bottom of the computational domain starting from x = 1 6 . Initially a right-moving Mach 10 shock is po... |

23 | C.-W.Shu A numerical resolution study of high order essentially nonoscillatory schemes applied to incompressible flow
- E
- 1993
(Show Context)
Citation Context ... compressible turbulence [71], [80], [49]; to relativistic hydrodynamics equations [24]; to shock vortex interactions and other gas dynamics problems [12], [27], [43]; to incompressible flow problems =-=[26]-=-, [31]; to viscoelasticity equations with fading memory [72]; to semi-conductor device simulation [28], [41], [42]; to image processing [59], [64], [73]; etc. This list is definitely incomplete and ma... |

20 |
resolution schemes using °ux limiters for hyperbolic conservative laws
- 15Sweby, \High
- 1984
(Show Context)
Citation Context ...ue of time discretization. The techniques discussed in this section can also be applied to other types of spatial discretizations using the method of lines approach, suchasvarious TVD and TVB schemes =-=[52, 78, 65]-=- and discontinuous Galerkin methods [18, 19, 20, 21, 17]. 4.2.1 TVD Runge-Kutta methods. A class of TVD (total variation diminishing) high order Runge-Kutta methods is developed in [69] and further in... |

19 |
An Eulerian approach for vortex motion using a level set regularization procedure
- Harabetian, Osher, et al.
- 1996
(Show Context)
Citation Context ...essible turbulence [71], [80], [49]; to relativistic hydrodynamics equations [24]; to shock vortex interactions and other gas dynamics problems [12], [27], [43]; to incompressible flow problems [26], =-=[31]-=-; to viscoelasticity equations with fading memory [72]; to semi-conductor device simulation [28], [41], [42]; to image processing [59], [64], [73]; etc. This list is definitely incomplete and may be b... |

19 | On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions.
- Iske, Sonar
- 1996
(Show Context)
Citation Context ... developed, using a convex combination of all candidate stencils instead of just one as in the original ENO, [53], [43]. ENO schemes based on other than polynomial building blocks were constructed in =-=[40]-=-, [16]. Sub-cell resolution and artificial compression to sharpen contact discontinuities were studied in [35], [83], [70] and [43]. Multidimensional ENO schemes based on general triangulation were de... |

19 |
A numerical study of the convergence properties of ENO schemes
- Rogerson, Meiburg
(Show Context)
Citation Context ...ion of the inequality and hence the stencil. In smooth regions, this "free adaptation" of stencils is clearly not necessary. Moreover, this may cause loss of accuracy when applied to a hyper=-=bolic PDE [63, 67]-=-. 2. The resulting numerical flux (2.23) is not smooth, as the stencil pattern may change at neighboring points. 3. In the stencil choosing process, k candidate stencils are considered, covering 2k - ... |

18 |
Riemann solvers, the entropy condition, and dierence approximations
- Osher
- 1984
(Show Context)
Citation Context ...) and interchanging the order of the two ext's in (4.42) can produce a di#erent monotone Hamiltonian. Godunov Hamiltonian is purely upwind and is the least dissipative among all monotone Hamiltonians =-=[57]-=-. However, it might be extremely di#cult to program, since in general analytical expressions for things like min u max v H(u, v) can be quite complicated. The readers will be convinced by doing the ex... |

17 |
Some Results on Uniformly High Order Accurate Essentially Non-oscillatory Schemes
- Harten, Osher, et al.
- 1986
(Show Context)
Citation Context ... smooth function V (x), ENO interpolation starting with a two point stencil S 2 (i)= {x i1 2 ,x i+ 1 2 } in the Step 2 of Procedure 2.1, as was shown in Fig. 2.1 (right), has the following properties =-=[39]-=-: 1. The accuracy condition P i (x)=V (x)+O(#x k+1 ),x# I i is valid for any cell I i which does not contain a discontinuity. This implies that the ENO interpolation procedure can recover the full hig... |

16 | Comparison of two formulations for high-order accurate essentially nonoscillatory schemes
- Casper, Shu, et al.
- 1994
(Show Context)
Citation Context ...on here is also valid for higher spatial dimension n. In e#ect, it is the same one dimensional conservative derivative approximation applied to each space dimension. It is a straight forward exercise =-=[13]-=- to show that, in terms of operation count, the finite di#erence ENO or WENO schemes are about a factor of 4 less than the finite volume counterpart of the same order. In 3D this factor becomes about ... |

15 | Sundström A. Stability theory of dierence approximations for mixed initial boundary value problems - Gustafsson, HO - 1972 |

15 |
Shock Waves and Reaction Diusion Equations
- Smoller
- 1983
(Show Context)
Citation Context ...gas, whichisusedvery often in practice. However, it is usually very costly to get this solution (for Euler equations of compressible gas, an iterative procedure is needed to obtain this solution, see =-=[74]-=-). In practice, approximate Riemann solvers are usually good enough. As in the scalar case, the quality of the solution is usually very sensitive to the choice of approximate Riemann solvers for lower... |

14 |
An Artificial Compression Method for ENO schemes: the SLOpe Modification Method
- Yang
- 1990
(Show Context)
Citation Context ...43]. ENO schemes based on other than polynomial building blocks were constructed in [40], [16]. Sub-cell resolution and artificial compression to sharpen contact discontinuities were studied in [35], =-=[83]-=-, [70] and [43]. Multidimensional ENO schemes based on general triangulation were developed in [1]. ENO and WENO schemes for Hamilton-Jacobi type equations were designed and applied in [59], [60], [50... |

13 |
H.L.: A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems.
- Caspr, Atkins
- 1993
(Show Context)
Citation Context ...ons [70], [71], [2]; to the direct 2 simulation of compressible turbulence [71], [80], [49]; to relativistic hydrodynamics equations [24]; to shock vortex interactions and other gas dynamics problems =-=[12]-=-, [27], [43]; to incompressible flow problems [26], [31]; to viscoelasticity equations with fading memory [72]; to semi-conductor device simulation [28], [41], [42]; to image processing [59], [64], [7... |

13 |
High Order ENO Schemes Applied to Two- and Three
- Shu, Zang, et al.
- 1992
(Show Context)
Citation Context ...NO with spectral method using a domain decomposition approach was carried out in [8]. On the application side, ENO and WENO have been successfully used to simulate shock turbulence interactions [70], =-=[71]-=-, [2]; to the direct 2 simulation of compressible turbulence [71], [80], [49]; to relativistic hydrodynamics equations [24]; to shock vortex interactions and other gas dynamics problems [12], [27], [4... |

12 | Transport effects and characteristic modes in the modeling and simulation of submicron devices - Jerome, Shu - 1995 |

11 |
Relativistic Hydrodynamics and Essentially Non-Oscillatory Shock Capturing Schemes
- Dolezal, Wong
- 1995
(Show Context)
Citation Context ... have been successfully used to simulate shock turbulence interactions [70], [71], [2]; to the direct 2 simulation of compressible turbulence [71], [80], [49]; to relativistic hydrodynamics equations =-=[24]-=-; to shock vortex interactions and other gas dynamics problems [12], [27], [43]; to incompressible flow problems [26], [31]; to viscoelasticity equations with fading memory [72]; to semi-conductor dev... |

11 |
Preliminary results on the extension of ENO schemes to two-dimensional problems
- Harten
- 1986
(Show Context)
Citation Context ...als in n - 1 dimension and a n - 1 dimensional quadrature rule must be used. 39 This is why multidimensional finite volume schemes of order of accuracy higher than 2 are rarely used. For 2D, based on =-=[34]-=-, Casper [11] has coded up a fourth order finite volume ENO scheme for Cartesian grids, see also [12]. 3D finite volume ENO code of order of accuracy higher than 2 does not exist yet, to the author's ... |

10 |
Uniform high-order spectral methods for one- and twodimensional Euler equations
- Cai, Shu
- 1993
(Show Context)
Citation Context ... of systems, were discussed in [25]. Combination of ENO with multiresolution ideas was pursued in [7]. Combination of ENO with spectral method using a domain decomposition approach was carried out in =-=[8]-=-. On the application side, ENO and WENO have been successfully used to simulate shock turbulence interactions [70], [71], [2]; to the direct 2 simulation of compressible turbulence [71], [80], [49]; t... |

10 |
Initial boundary value problems for method of lines
- Strikwerda
- 1980
(Show Context)
Citation Context ...s actually equivalent to the approach of using only the stencils inside the computational domain in the ENO procedure. WENO can be handled in a similar fashion. Stability analysis (GKS analysis [30], =-=[76]-=-) can be used to study the linear stability when the boundary treatment described above is applied to a fixed stencil upwind biased scheme. For most practical situations the schemes are linearly stabl... |

9 |
Application of generalized wavelets: an adaptive multiresolution scheme
- Bihari, Harten
- 1995
(Show Context)
Citation Context ...g one-sided Jocobians for field by field decomposition, which improves the robustness for calculations of systems, were discussed in [25]. Combination of ENO with multiresolution ideas was pursued in =-=[7]-=-. Combination of ENO with spectral method using a domain decomposition approach was carried out in [8]. On the application side, ENO and WENO have been successfully used to simulate shock turbulence i... |

9 |
Finite-volume implementation of high-order essentially nonoscillatory schemes in two dimensions
- Casper
- 1992
(Show Context)
Citation Context ...dimension and a n - 1 dimensional quadrature rule must be used. 39 This is why multidimensional finite volume schemes of order of accuracy higher than 2 are rarely used. For 2D, based on [34], Casper =-=[11]-=- has coded up a fourth order finite volume ENO scheme for Cartesian grids, see also [12]. 3D finite volume ENO code of order of accuracy higher than 2 does not exist yet, to the author's knowledge. At... |

9 |
TVB Uniformly High Order Schemes for Conservation Laws
- Shu
- 1987
(Show Context)
Citation Context ... of time discretization. The techniques discussed in this section can also be applied to other types of spatial discretizations using the method of lines approach, such as various TVD and TVB schemes =-=[52, 78, 65]-=- and discontinuous Galerkin methods [18, 19, 20, 21, 17]. 4.2.1. TVD Runge-Kutta methods. A class of TVD (total variation diminishing) high order RungeKutta methods is developed in [69] and further in... |

9 |
Numerical analysis of blood °ow in the heart
- Peskin
- 1977
(Show Context)
Citation Context ...idal perturbation of a at sheet: '0(x� y) =y +0:05 sin( x) The boundary condition for ' are periodic, of the form: '(t� ;1�y)='(t� 1�y) '(t� x� ;1) = '(t� x� 1) ; 2 The function is approximated as in =-=[61]-=-,[77] by ( )= ( 1 2 (1 + cos ) if j'j < 0 otherwise 80 (5:32)s0.5 0.0 -0.5 t=4 128 2 points ε = 12 ∆ x -1.0 -1.0 -0.5 0.0 0.5 1.0 1.5 0.5 0.0 -0.5 t=4 -1.0 -1.0 -0.5 0.0 0.5 1.0 1.5 512 2 points ε = 2... |

9 |
The numerical simulation of two-dimensional uid ow with strong shocks
- PR, Colella
- 1984
(Show Context)
Citation Context ...e residue in this case settles down nicely to machine zeros. Both fourth and fth order WENO results are shown. 4. Forward facing step problem. This is a standard test case for high resolution schemes =-=[82]-=-. However, second order methods usually already work well. High order methods might have some advantage in resolving the slip lines. The set up of the problem is the following: the wind tunnel is 1 le... |

6 |
Advection by polytropic compressible turbulence
- Ladeinde, O'Brien, et al.
- 1995
(Show Context)
Citation Context ... in [8]. On the application side, ENO and WENO have been successfully used to simulate shock turbulence interactions [70], [71], [2]; to the direct 2 simulation of compressible turbulence [71], [80], =-=[49]-=-; to relativistic hydrodynamics equations [24]; to shock vortex interactions and other gas dynamics problems [12], [27], [43]; to incompressible flow problems [26], [31]; to viscoelasticity equations ... |

6 |
Robust numerical methods for 2D turbulence
- Walsteijn
- 1994
(Show Context)
Citation Context ...ed out in [8]. On the application side, ENO and WENO have been successfully used to simulate shock turbulence interactions [70], [71], [2]; to the direct 2 simulation of compressible turbulence [71], =-=[80]-=-, [49]; to relativistic hydrodynamics equations [24]; to shock vortex interactions and other gas dynamics problems [12], [27], [43]; to incompressible flow problems [26], [31]; to viscoelasticity equa... |

5 |
On dierence approximations and entropy conditions for shocks
- Harten, Hyman, et al.
- 1976
(Show Context)
Citation Context ...st 1 2 [23]. In the semi-discrete formulation, a five point monotone scheme (it does not pay to use more points for a monotone scheme because the order of accuracy of a monotone scheme is at most one =-=[36]-=-) is of the form d dt # ij (t)=- H(D x + # ij (t),D x - # ij (t),D y + # ij (t),D y - # ij (t)). (4.38) 50 The numerical HamiltoniansH is assumed to be locally Lipschitz continuous, consistent with H ... |

5 |
A practical assessment of spectral accuracy for hyperbolic problems with discontinuities
- Kopriva
- 1987
(Show Context)
Citation Context ... which, when used twice, will produce the fourth order central di#erence approximation 16(w i+1+w i-1 )-(w i+2+w i-2 )-30w i 12#x 2 for w xx . High order filters, such as the exponential filter [55], =-=[46]-=-: # x k = e -#( k Nx ) 2p ,# y l = e -#( l Ny ) 2p (5.14) where 2p is the order of the filter and # is chosen so that e -# is machine zero, can be used to enhance the stability while keeping at least ... |

5 | Discrete shocks for finite difference approximations to scalar conservation laws - Jiang, Yu - 1998 |

4 | Applicability of the high field model: An analytical study via symptotic parameters defining domain decomposition - Cercignani, Gamba, et al. - 1998 |

4 | Applicability of the high field model: a preliminary numerical study, VLSI Design - CERCIGNANI, GAMBA, et al. |

4 |
High-order 2-dimensional nonoscillatory methods for solving Hamilton-Jacobi scalar equations
- Lafon, Osher
- 1996
(Show Context)
Citation Context ...83], [70] and [43]. Multidimensional ENO schemes based on general triangulation were developed in [1]. ENO and WENO schemes for Hamilton-Jacobi type equations were designed and applied in [59], [60], =-=[50]-=- and [45]. ENO schemes using one-sided Jocobians for field by field decomposition, which improves the robustness for calculations of systems, were discussed in [25]. Combination of ENO with multiresol... |

4 |
The Arti cial Compression Method for Computation of Shocks and Contact Discontinuities
- Harten
- 1979
(Show Context)
Citation Context ...ocessing problems (we termed it geometric ENO, or GENO). 4.1.2 Arti cial compression. Another very useful idea to sharpen a contact discontinuity is the arti cial compression, rst developed by Harten =-=[32]-=- and further improved by Yang [83]. The idea is to increase the magnitude of the slope of a reconstruction, of course subject to certain monotonicity restrictions, near such a discontinuity. Notice th... |

4 | Numerical experiments on the accuracy of ENO and modi ed ENO schemes - Shu - 1990 |

3 |
GKS and eigenvalue stability analysis of high order upwind scheme
- Atkins, Shu
(Show Context)
Citation Context ...can be used to study the linear stability when the boundary treatment described above is applied to a fixed stencil upwind biased scheme. For most practical situations the schemes are linearly stable =-=[3]-=-. 2.3.4. Provable properties in the scalar case. Second order ENO schemes are also TVD (total variation diminishing), hence have at least subsequences which converge to weak solutions. There is no kno... |

3 |
The study of building blocks for ENO schemes
- Christofi
- 1995
(Show Context)
Citation Context ...oped, using a convex combination of all candidate stencils instead of just one as in the original ENO, [53], [43]. ENO schemes based on other than polynomial building blocks were constructed in [40], =-=[16]-=-. Sub-cell resolution and artificial compression to sharpen contact discontinuities were studied in [35], [83], [70] and [43]. Multidimensional ENO schemes based on general triangulation were develope... |

3 |
Transport e#ects and characteristic modes in the modeling and simulation of submicron devices
- Jerome, Shu
- 1995
(Show Context)
Citation Context ...ons and other gas dynamics problems [12], [27], [43]; to incompressible flow problems [26], [31]; to viscoelasticity equations with fading memory [72]; to semi-conductor device simulation [28], [41], =-=[42]-=-; to image processing [59], [64], [73]; etc. This list is definitely incomplete and may be biased by the author's own research experience, but one can already see that ENO and WENO have been applied q... |

3 |
L.Roe.Approximate Riemann solver, parameter vectors, and di erence schemes
- unknown authors
- 1981
(Show Context)
Citation Context ...use v ; i+ 1 2 the numerical ux ^ f i+ 1 2 � if a i+ 1 2 < 0, then the wind blows from the right to the left. We would use v+ i+ 1 2 numerical ux ^ f i+ 1 2 . for for the This produces the Roe scheme =-=[62]-=- at the rst order level. For this reason, the ENO scheme based on this approach was termed \ENO-Roe" in [70]. In summary, to build a nite di erence ENO scheme (2.73) using the ENO-Roe approach, given ... |

3 |
Preface to the republication of "Uniform high order essentially non-oscillatory schemes, III
- Shu
- 1997
(Show Context)
Citation Context ...at least 144 times by early 1997, according to the ISI database. The Journal of Computational Physics decided to republish this classic paper as part of the celebration of the journal's 30th birthday =-=[68]-=-. Finite di erence and related nite volume schemes are based on interpolations of discrete data using polynomials or other simple functions. In the approximation theory, itiswell known that the wider ... |

2 |
Capturing shock re ections: An improved ux formula
- Donat, Marquina
- 1996
(Show Context)
Citation Context ...ere designed and applied in [59], [60], [50] and [45]. ENO schemes using one-sided Jocobians for eld by eld decomposition, which improves the robustness for calculations of systems, were discussed in =-=[25]-=-. Combination of ENO with multiresolution ideas was pursued in [7]. Combination of ENO with spectral method using a domain decomposition approach was carried out in [8]. On the application side, ENO a... |

2 |
An arti cial compression method for ENO schemes, the slope modi cation method
- Yang
- 1990
(Show Context)
Citation Context ...[43]. ENO schemes based on other than polynomial building blocks were constructed in [40], [16]. Sub-cell resolution and arti cial compression to sharpen contact discontinuities were studied in [35], =-=[83]-=-, [70] and [43]. Multidimensional ENO schemes based on general triangulation were developed in [1]. ENO and WENO schemes for Hamilton-Jacobi type equations were designed and applied in [59], [60], [50... |

1 |
Discontinuous Galerkin method, this volume
- Cockburn
(Show Context)
Citation Context ...d in this section can also be applied to other types of spatial discretizations using the method of lines approach, such as various TVD and TVB schemes [52, 78, 65] and discontinuous Galerkin methods =-=[18, 19, 20, 21, 17]-=-. 4.2.1. TVD Runge-Kutta methods. A class of TVD (total variation diminishing) high order RungeKutta methods is developed in [69] and further in [29]. These Runge-Kutta methods are used to solve a sys... |

1 |
Discrete shocks for finite di#erence approximations to scalar conservation laws
- Jiang, Yu
(Show Context)
Citation Context ...ooth solutions. WENO schemes have better convergence results, mainly because their numerical fluxes are smoother. It is proven [43] that WENO schemes converge for smooth solutions. Also, Jiang and Yu =-=[44]-=- have obtained an existence proof for traveling waves for WENO schemes. This is an important first step towards the proof of convergence for shocked cases. Even though there are very little theoretica... |

1 |
The Fourier Method for Nonsmooth Initial
- Majda, McDonough, et al.
- 1978
(Show Context)
Citation Context ...rences which, when used twice, will produce the fourth order central di#erence approximation 16(w i+1+w i-1 )-(w i+2+w i-2 )-30w i 12#x 2 for w xx . High order filters, such as the exponential filter =-=[55]-=-, [46]: # x k = e -#( k Nx ) 2p ,# y l = e -#( l Ny ) 2p (5.14) where 2p is the order of the filter and # is chosen so that e -# is machine zero, can be used to enhance the stability while keeping at ... |

1 |
Preface to the republication of "Uniform high order essentially non-oscillatory schemes, III
- Shu
- 1997
(Show Context)
Citation Context ...at least 144 times by early 1997, according to the ISI database. The Journal of Computational Physics decided to republish this classic paper as part of the celebration of the journal's 30th birthday =-=[68]-=-. Finite di#erence and related finite volume schemes are based on interpolations of discrete data using polynomials or other simple functions. In the approximation theory, it is well known that the wi... |

1 | Applicability of the high eld model: an analytical study via asymptotic parameters de ning domain decomposition, VLSI Design - Cercignani, Gamba, et al. |

1 | Applicability of the high eld model: apreliminary numerical study, VLSI Design - Cercignani, Gamba, et al. |

1 |
The study of building blocks for ENO schemes
- Christo
- 1995
(Show Context)
Citation Context ...oped, using a convex combination of all candidate stencils instead of just one as in the original ENO, [53], [43]. ENO schemes based on other than polynomial building blocks were constructed in [40], =-=[16]-=-. Sub-cell resolution and arti cial compression to sharpen contact discontinuities were studied in [35], [83], [70] and [43]. Multidimensional ENO schemes based on general triangulation were developed... |

1 |
Discontinuous Galerkin method, thisvolume
- Cockburn
(Show Context)
Citation Context ...sed in this section can also be applied to other types of spatial discretizations using the method of lines approach, suchasvarious TVD and TVB schemes [52, 78, 65] and discontinuous Galerkin methods =-=[18, 19, 20, 21, 17]-=-. 4.2.1 TVD Runge-Kutta methods. A class of TVD (total variation diminishing) high order Runge-Kutta methods is developed in [69] and further in [29]. These Runge-Kutta methods are used to solve a sys... |

1 |
Transport e ects and characteristic modes in the modeling and simulation of submicron devices
- Jerome, Shu
- 1995
(Show Context)
Citation Context ...tions and other gas dynamics problems [12], [27], [43]� to incompressible ow problems [26], [31]� to viscoelasticity equations with fading memory [72]� to semi-conductor device simulation [28], [41], =-=[42]-=-� to image processing [59], [64], [73]� etc. This list is de nitely incomplete and may be biased by the author's own research experience, but one can already see that ENO and WENO have been applied qu... |

1 |
Discrete shocks for nite di erence approximations to scalar conservation laws
- Jiang, Yu
(Show Context)
Citation Context ...smooth solutions. WENO schemes have better convergence results, mainly because their numerical uxes are smoother. It is proven [43] that WENO schemes converge for smooth solutions. Also, Jiang and Yu =-=[44]-=-have obtained an existence proof for traveling waves for WENO schemes. This is an important rst step towards the proof of convergence for shocked cases. Even though there are very little theoretical r... |