yingjie liu school of math, georgia tech joint work with: zhiliang xu and chi-wang shu
DESCRIPTION
Non-Oscillatory Hierarchical Reconstruction for DG on Triangular Meshes with a WENO-type Linear Reconstruction. Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu. This work is the further development of the following two papers. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/1.jpg)
Non-Oscillatory Hierarchical Reconstruction for DG on
Triangular Meshes with a WENO-type Linear Reconstruction
• Yingjie Liu
• School of Math, Georgia Tech
• Joint work with:
• Zhiliang Xu and Chi-Wang Shu
![Page 2: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/2.jpg)
• This work is the further development of the following two papers.
• Liu, Shu, Tadmor and Zhang, on hierarchical reconstruction for central DG, SINUM ’07.
• ___, on hierarchical reconstruction for central and finite volume schemes, CiCP ’07.
![Page 3: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/3.jpg)
![Page 4: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/4.jpg)
![Page 5: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/5.jpg)
![Page 6: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/6.jpg)
![Page 7: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/7.jpg)
![Page 8: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/8.jpg)
![Page 9: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/9.jpg)
![Page 10: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/10.jpg)
Remarks
• For multi dimensions, partial derivatives of all orders need to be taken.
• The key is to use the (updated) higher degree remainder to estimate cell averages of the target linear part over neighboring cells, then on each hierarchy use a multi-D MUSCL-type reconstruction to reconstruct the linear part.
• Because the linear reconstruction on each hierarchy is compact and applicable to any mesh structure or cell shape, so is the hierarchical reconstruction in theory.
![Page 11: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/11.jpg)
Further Remarks
• Hierarchical reconstruction does not change the approximation order of the polynomial.
• It does not need characteristic decomposition.• On each hierarchy, given cell averages on
neighboring cells, one can also use a weighted average of the linear reconstructions—the WENO strategy(indep.of local mesh), which builds a smoother shift among stencils.
• It increases the CFL numbers to about 0.5 for P2 DG.
![Page 12: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/12.jpg)
Hierarchical reconstruction only uses adjacent cells for any order
![Page 13: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/13.jpg)
Triangular mesh
![Page 14: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/14.jpg)
![Page 15: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/15.jpg)
![Page 16: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/16.jpg)
![Page 17: Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu](https://reader035.vdocuments.us/reader035/viewer/2022062409/56814f5c550346895dbd0cd6/html5/thumbnails/17.jpg)
P2 DG. Triangle edge length=1/300