Improved Five- and Six-Point Targeted Essentially Nonoscillatory Schemes with Adaptive Dissipation

AIAA Journal ◽  
2019 ◽  
Vol 57 (3) ◽  
pp. 1143-1158 ◽  
Author(s):  
Lin Fu ◽  
Xiangyu Y. Hu ◽  
Nikolaus A. Adams
Keyword(s):  
Essentially Nonoscillatory

scholarly journals Entropy stable essentially nonoscillatory methods based on RBF reconstruction

2019 ◽  
Vol 53 (3) ◽  
pp. 925-958 ◽  
Author(s):  
Jan S. Hesthaven ◽  
Fabian Mönkeberg
Keyword(s):  
Radial Basis Functions ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Basis Functions ◽  
One Dimensional ◽  
Hyperbolic Conservation ◽  
Radial Basis ◽  
Sign Property ◽  
Entropy Stable ◽  
Essentially Nonoscillatory

To solve hyperbolic conservation laws we propose to use high-order essentially nonoscillatory methods based on radial basis functions. We introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Thus, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. The main contribution is the construction of an algorithm and a smoothness indicator that ensures an interpolation function which fulfills the sign-property on general one dimensional grids.

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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