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