## Efficient Implementation of Weighted ENO Schemes (1995)

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Citations: | 243 - 27 self |

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@MISC{Jiang95efficientimplementation,

author = {Guang-shan Jiang and Chi-wang Shu},

title = {Efficient Implementation of Weighted ENO Schemes },

year = {1995}

}

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### Abstract

In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially non-oscillatory) finite difference schemes of Liu, Osher and Chan [9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L¹ norm) ENO schemes are (r +1) th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimizing the total variation of the approximation, which results in a 5th order WENO scheme for the case r = 3, instead of the 4th order with the original smoothness measurement by Liu et al. This 5 th order WENO scheme is as fast as the 4 th order WENO scheme of Liu et al. and, both schemes are about twice as fast as the 4th order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest to compute the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights, and work well for problems which donot contain strong shocks or strong reflected waves. We also prove that, for conservation laws with smooth solutions, all WENO schemes are convergent. Many numerical tests, including the 1D steady state nozzle flow problem and 2D shock entropy waveinteraction problem, are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement, in resolving complicated shock and flow structures. We have also applied Yang's artificial compression method to the WENO schemes to sharpen contact discontinuities.

### Citations

622 | Approximate Riemann solver, parameters vectors and difference schemes
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589 |
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- 1988
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Citation Context ...ell-averaged version of ENO schemes involves a procedure of reconstructing point values from cell averages and could become complicated and costly for multi-dimensional problems. Later, Shu and Osher =-=[14, 15]-=- developed the flux version of ENO schemes which do not require such a reconstruction procedure. We will formulate the WENO schemes based on this flux version of ENO schemes. The WENO schemes of Liu e... |

189 | Weighted essentially non-oscillatory schemes
- Liu, Osher, et al.
- 1994
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Citation Context ...ce, Rhode Island 02912 Abstract In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially non-oscillatory) finite difference schemes of Liu, Osher and Chan =-=[9]-=-. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L 1 norm) ENO schemes are (r +1) th order accurate. We propose a new way of measuring the smoothness of a numerical s... |

169 |
The numerical simulation of two-dimensional fluid flow with strong shocks
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- 1984
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127 |
Weak solutions of nonlinear hyperbolic equations and their numerical calculation
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- 1954
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Citation Context ...): WENO-RF-5. The first one is the Sod's problem [17]. The initial data are: (ae L ; q L ; PL ) = (1; 0; 1); (ae R ; q R ; PR ) = (0:125; 0; 0:1) The second one is the Riemann problem proposed by Lax =-=[7]-=-: (ae L ; q L ; PL ) = (0:445; 0:698; 3:528); (ae R ; q R ; PR ) = (0:5; 0; 0:571) The numerical results are presented in Figure 4. We can see that all schemes give correct solution with good resoluti... |

116 |
A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws
- Sod
- 1978
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Citation Context ...-3 (b) -1.0-0.5 0.0 0.5 1.0 -0.0 0.2 0.4 0.6 0.8 1.0 o WENO-RF-5 �� EXACT t=0.4 cfl=0.8 n=81 Figure 3: The Buckley-Leverett problem. (a): WENO-RF-3; (b): WENO-RF-5. The first one is the Sod's prob=-=lem [17]-=-. The initial data are: (ae L ; q L ; PL ) = (1; 0; 1); (ae R ; q R ; PR ) = (0:125; 0; 0:1) The second one is the Riemann problem proposed by Lax [7]: (ae L ; q L ; PL ) = (0:445; 0:698; 3:528); (ae ... |

67 |
Uniform high-order accurate essentially non-oscillatory schemes
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- 1997
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Citation Context ... (f 1 ; : : : ; f d ); x = (x 1 ; : : : ; x d ) and t ? 0: WENO schemes are based on ENO (essentially non-oscillatory) schemes, which were first introduced by Harten, Osher, Engquist and Chakravarthy =-=[5] in the fo-=-rm of cell averages. The key idea of ENO schemes is to use the "smoothest" stencil among several candidates to approximate the fluxes at cell boundaries to a high order accuracy and at the s... |

51 |
schemes with subcell resolution
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- 1989
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Citation Context ...chemes have the same smearing at contact discontinuities as ENO schemes. There are mainly two techniques for sharpening the contact discontinuities for ENO schemes. One is Harten's subcell resolution =-=[4]-=- and the other is Yang's artificial compression (slope modification) [20]. Both were introduced in the cell average context. Later, Shu and Osher [15] translated them into the point value framework. I... |

38 |
Numerical experiments on the accuracy of ENO and modified ENO schemes
- Shu
- 1990
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Citation Context ...died by embedding certain parameters (e.g. threshold and biasing factor) into the stencil choosing step so that the preferred linearly stable stencil is used in regions away from discontinuities. See =-=[1, 3, 13]-=-. WENO scheme of Liu, Osher and Chan [9] is another way to overcome these drawbacks while keeping the robustness and high order accuracy of ENO schemes. The idea is the following: instead of approxima... |

24 |
Accurate partial difference methods II: Nonlinear problems
- Strang
- 1964
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Citation Context ...ther advantage of WENO schemes is that its flux is smoother than that of the ENO schemes. This smoothness enables us to prove convergence of WENO schemes for smooth solutions using Strang's technique =-=[18]-=-, see Section 6. According to our numerical tests, this smoothness also helps the steady state calculations, see Example 4 in Section 8.2. In [9], the order of accuracy shown in the error tables (Tabl... |

13 |
A numerical study of the convergence properties of ENO schemes
- Rogerson, Meiburg
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Citation Context ...0 because the stencil choosing step involves heavy usage of logical statements, which perform poorly on such machines. The first problem could reduce the accuracy of ENO schemes for certain functions =-=[12]-=-, however this can be remedied by embedding certain parameters (e.g. threshold and biasing factor) into the stencil choosing step so that the preferred linearly stable stencil is used in regions away ... |

11 |
Comparison of two formulations for high-order accurate essentially nonoscillatory schemes
- Casper, Shu, et al.
- 1994
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Citation Context ...died by embedding certain parameters (e.g. threshold and biasing factor) into the stencil choosing step so that the preferred linearly stable stencil is used in regions away from discontinuities. See =-=[1, 3, 13]-=-. WENO scheme of Liu, Osher and Chan [9] is another way to overcome these drawbacks while keeping the robustness and high order accuracy of ENO schemes. The idea is the following: instead of approxima... |

11 |
An Artificial Compression Method for ENO schemes: the SLOpe Modification Method
- Yang
- 1990
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Citation Context ... There are mainly two techniques for sharpening the contact discontinuities for ENO schemes. One is Harten's subcell resolution [4] and the other is Yang's artificial compression (slope modification) =-=[20]-=-. Both were introduced in the cell average context. Later, Shu and Osher [15] translated them into the point value framework. In one dimensional problems, subcell resolution technique works slightly b... |

10 |
High-Order ENO Schemes Applied to Two and Three Dimensional Compressible Flow
- Shu
- 1991
(Show Context)
Citation Context ...raction are much more difficult to resolve than in the 1D case. Our goal here is to further examine the capability of the WENO scheme in capturing such small scale waves at the presence of shock. See =-=[21, 16]-=- for detailed discussions on this subject. The set-up of the problem is the following: for a right moving normal shock of Mach number M , we add a small entropy wave to the flow on the right of the sh... |

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Interaction of linear waves with oblique shock waves, Phys
- Mckenzie, Westphal
- 1968
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Citation Context ...ENO-LF-5-PS (h) -4-2 0 2 4 0.4 0.6 0.8 1.0 1.2 DENSITY t=1.3 cfl=0.6 n=101 �� EXACT o WENO-LF-5-A Figure 4: Density. (a)-(d). Sod's problem. (e)-(h) Lax's problem. be obtained by linear analysis. =-=See [10, 21]-=- for details. In order to get rid of the transient waves due to the non-numerical initial shock profile, we let the shock move up to x = 4:5 and then shuffle it back to x = 0:5. The solution is examin... |

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Finite-volume implementation of high-order essentially non-oscillatory schemes in two dimensions
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- 1992
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Citation Context ...look at the following model problem. Example 5. Shock Vortex Interaction. This model problem describes the interaction between a stationary shock and a vortex. The computational domain is taken to be =-=[0; 2]-=- \Theta [0; 1]. A stationary Mach 1.1 shock is positioned at x = 0:5 and normal to the x-axis. Its left state is (ae; u; v; P ) = (1; p fl; 0; 1). A small vortex is superposed to the flow left to the ... |

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High-Order ENO Methods for the Unsteady Compressible Navier-Stokes Equations
- Atkins
- 1991
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Citation Context ...died by embedding certain parameters (e.g. threshold and biasing factor) into the stencil choosing step so that the preferred linearly stable stencil is used in regions away from discontinuities. See =-=[1, 3, 13]-=-. WENO scheme of Liu, Osher and Chan [9] is another way to overcome these drawbacks while keeping the robustness and high order accuracy of ENO schemes. The idea is the following: instead of approxima... |

5 |
Numerical computations of turbulence amplification in shock-wave interactions
- Zang, Hussaini, et al.
- 1984
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Citation Context ...ENO-LF-5-PS (h) -4-2 0 2 4 0.4 0.6 0.8 1.0 1.2 DENSITY t=1.3 cfl=0.6 n=101 �� EXACT o WENO-LF-5-A Figure 4: Density. (a)-(d). Sod's problem. (e)-(h) Lax's problem. be obtained by linear analysis. =-=See [10, 21]-=- for details. In order to get rid of the transient waves due to the non-numerical initial shock profile, we let the shock move up to x = 4:5 and then shuffle it back to x = 0:5. The solution is examin... |

5 | A survey of several ¯nite di®erence methods for systems of nonlinear hyperbolic conservation laws - Sod - 1978 |

5 | Accurate partial dierence methods. II: Non-linear problems - Strang - 1964 |

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

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Positive schemes for solving multi-dimensional systems of hyperbolic conservation laws
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Citation Context ...a bit slower on SGI Indigo2. WENO-LF-5-PS is 1.5 to 2.2 times as fast as ENO-LF-4 on SUN Sparc10 and on SGI Indigo2. As a reference, we also include the CPU times of a typical second order TVD scheme =-=[8]-=- (Van Leer's limiter with 2 nd order Runge-Kutta scheme in time, our own implementation) in the following tables. We can see the 2 nd order scheme is about 10 times as fast as ENO-LF-4 on CRAY C-90, 4... |

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Algorithm analysis and efficient computation of conservation laws
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- 1995
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Citation Context ...`s�� 2 ). If ` = �� 2 , we have the one dimension Sod's problem. If 0 ! ` ! �� 2 , all the waves produced will be oblique to the rectangular computational mesh. We take our computational d=-=omain to be [0; 6]-=- \Theta [0; 1] and position the initial jump at (x; y) = (2:25; 0). The physical domain varies with ` and is taken as [0; 6 sin ` ] \Theta [0; 1 sin ` ]. The scaling factor 1 sin ` is to ensure the sa... |

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