Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells

JOURNAL OF COMPUTATIONAL PHYSICS

卷:228

Article

Xu, Zhiliang¹  Liu, Yingjie²  Shu, Chi-Wang³ 

[1] Univ Notre Dame, Dept Math, Notre Dame, IN 46556 USA

[2] Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA

[3] Brown Univ, Div Appl Math, Providence, RI 02912 USA

关键词: Hierarchical reconstruction;    Discontinuous Galerkin methods;    Unstructured grids;    Hyperbolic conservation laws;   

DOI  :  10.1016/j.jcp.2008.11.025

来源: Elsevier

【 摘 要 】

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442-2467] is applied to the piecewise quadratic discontinuous Galerkin method on two-dimensional unstructured triangular grids. A variety of limiter functions have been explored in the construction of piecewise linear polynomials in every hierarchical reconstruction stage. We show that on triangular grids, the use of center biased limiter functions is essential in order to recover the desired order of accuracy. Several new techniques have been developed in the paper: (a) we develop a WENO-type linear reconstruction in each hierarchical level, which solves the accuracy degeneracy problem of previous limiter functions and is essentially independent of the local mesh structure: (b) we find that HR using partial neighboring cells significantly reduces over/under-shoots. and further improves the resolution of the numerical solutions. The method is compact and therefore easy to implement. Numerical computations for scalar and systems of nonlinear hyperbolic equations are performed. We demonstrate that the procedure can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

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