Hierarchical reconstruction for spectral volume method on unstructured grids

  • Authors:

  • Zhiliang Xu;Yingjie Liu;Chi-Wang Shu

  • Affiliations:

  • Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States;Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2009

Quantified Score

Hi-index 31.46

Visualization

Abstract

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; Z.-L. Xu, Y.-J. Liu, C.-W. Shu, Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells, J. Comput. Phys. 228 (2009) 2194-2212] is applied to a piecewise quadratic spectral volume method on two-dimensional unstructured grids as a limiting procedure to prevent spurious oscillations in numerical solutions. The key features of this HR are that the reconstruction on each control volume only uses adjacent control volumes, which forms a compact stencil set, and there is no truncation of higher degree terms of the polynomial. We explore a WENO-type linear reconstruction on each hierarchical level for the reconstruction of high degree polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed. We demonstrate that the hierarchical reconstruction can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.