L 1 ‐type smoothness indicators based weighted essentially nonoscillatory scheme for Hamilton‐Jacobi equations

2020 ◽  
Vol 92 (12) ◽  
pp. 1927-1947 ◽  
Author(s):  
Samala Rathan
Keyword(s):  
Weighted Essentially Nonoscillatory ◽  
Smoothness Indicators ◽  
Jacobi Equations ◽  
Essentially Nonoscillatory

A new type of finite difference WENO schemes for Hamilton–Jacobi equations

2019 ◽  
Vol 30 (02n03) ◽  
pp. 1950020 ◽  
Author(s):  
Xiaohan Cheng ◽  
Jianhu Feng ◽  
Supei Zheng ◽  
Xueli Song
Keyword(s):  
Finite Difference ◽  
Weno Schemes ◽  
Numerical Examples ◽  
Weighted Essentially Nonoscillatory ◽  
New Type ◽  
Spurious Oscillations ◽  
Symmetry Requirement ◽  
Finite Difference Weno Schemes ◽  
Jacobi Equations ◽  
Fifth Order

In this paper, we propose a new type of finite difference weighted essentially nonoscillatory (WENO) schemes to approximate the viscosity solutions of the Hamilton–Jacobi equations. The new scheme has three properties: (1) the scheme is fifth-order accurate in smooth regions while keep sharp discontinuous transitions with no spurious oscillations near discontinuities; (2) the linear weights can be any positive numbers with the symmetry requirement and that their sum equals one; (3) the scheme can avoid the clipping of extrema. Extensive numerical examples are provided to demonstrate the accuracy and the robustness of the proposed scheme.

scholarly journals Fifth-Order Mapped Semi-Lagrangian Weighted Essentially Nonoscillatory Methods Near Certain Smooth Extrema

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Lang Wu ◽  
Dazhi Zhang ◽  
Boying Wu ◽  
Xiong Meng
Keyword(s):  
Finite Difference ◽  
Finite Volume ◽  
Interpolation Scheme ◽  
Numerical Examples ◽  
Weighted Essentially Nonoscillatory ◽  
Accuracy Test ◽  
A Cell ◽  
Fifth Order ◽  
Cell Average ◽  
Essentially Nonoscillatory

Fifth-order mapped semi-Lagrangian weighted essentially nonoscillatory (WENO) methods at certain smooth extrema are developed in this study. The schemes contain the mapped semi-Lagrangian finite volume (M-SL-FV) WENO 5 method and the mapped compact semi-Lagrangian finite difference (M-C-SL-FD) WENO 5 method. The weights in the more common scheme lose accuracy at certain smooth extrema. We introduce mapped weighting to handle the problem. In general, a cell average is applied to construct the M-SL-FV WENO 5 reconstruction, and the M-C-SL-FD WENO 5 interpolation scheme is proposed based on an interpolation approach. An accuracy test and numerical examples are used to demonstrate that the two schemes reduce the loss of accuracy and improve the ability to capture discontinuities.

scholarly journals Weighted Essentially Nonoscillatory Method for Two-Dimensional Population Balance Equations in Crystallization

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Chunlei Ruan ◽  
Kunfeng Liang ◽  
Xianjie Chang ◽  
Ling Zhang
Keyword(s):  
Crystal Size ◽  
Hyperbolic Equations ◽  
Population Balance ◽  
Numerical Dispersion ◽  
Two Dimensional ◽  
Governing Equations ◽  
Balance Equations ◽  
Population Balance Equations ◽  
Weighted Essentially Nonoscillatory ◽  
Essentially Nonoscillatory

Population balance equations (PBEs) are the main governing equations to model the processes of crystallization. Two-dimensional PBEs refer to the crystals that grow anisotropically with the change of two internal coordinates. Since the PBEs are hyperbolic equations, it is necessary to build up high resolution schemes to avoid numerical diffusion and numerical dispersion in order to obtain the accurate crystal size distribution (CSD). In this work, a 5th order weighted essentially nonoscillatory (WENO) method is introduced to compute the two-dimensional PBEs. Several numerical benchmark examples from literatures are carried out; it is found that WENO method has higher resolution than HR method which is well established. Therefore, WENO method is recommended in crystallization simulation when the crystal size distributions are sharp and higher accuracy is needed.

DEVELOPMENT AND ASSESSMENT OF SEVERAL HIGH-RESOLUTION SCHEMES FOR COMPRESSIBLE EULER EQUATIONS

2013 ◽  
Vol 11 (01) ◽  
pp. 1350049
Author(s):  
M. P. RAY ◽  
B. P. PURANIK ◽  
U. V. BHANDARKAR
Keyword(s):  
High Resolution ◽  
Temporal Integration ◽  
Splitting Scheme ◽  
Flux Vector ◽  
Riemann Solvers ◽  
Weighted Essentially Nonoscillatory ◽  
Time Stepping ◽  
Flux Vector Splitting ◽  
Piecewise Parabolic ◽  
Essentially Nonoscillatory

High-resolution extensions to six Riemann solvers and three flux vector splitting schemes are developed within the framework of a reconstruction-evolution approach. Third-order spatial accuracy is achieved using two different piecewise parabolic reconstructions and a weighted essentially nonoscillatory scheme. A three-stage TVD Runge–Kutta time stepping is employed for temporal integration. The modular development of solvers provides an ease in selecting a reconstruction scheme and/or a Riemann solver/flux vector splitting scheme. The performances of these high-resolution solvers are compared for several one- and two-dimensional test cases. Based on a comprehensive assessment of the solutions obtained with all solvers, it is found that the use of the weighted essentially nonoscillatory reconstruction with the van Leer flux vector splitting scheme provides solutions for a variety of problems with acceptable accuracy.

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