Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu
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Transcript of Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu
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.
• Liu, Shu, Tadmor and Zhang, on hierarchical reconstruction for central DG, SINUM ’07.
• ___, on hierarchical reconstruction for central and finite volume schemes, CiCP ’07.
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.
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.
Hierarchical reconstruction only uses adjacent cells for any order
Triangular mesh
P2 DG. Triangle edge length=1/300