Yingjie Liu School of Math, Georgia Tech Joint work with: Zhiliang Xu and Chi-Wang Shu

1. 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

2. 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.

10. 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.

11. 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.

12. Hierarchical reconstruction only uses adjacent cells for any order

13. Triangular mesh

17. P2 DG. Triangle edge length=1/300