Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

  • Authors:

  • Zhiliang Xu;Yingjie Liu;Huijing Du;Guang Lin;Chi-Wang Shu

  • Affiliations:

  • Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, United States;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States;Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, United States;Computational Mathematics Group, Pacific Northwest National Laboratory, 902 Battelle Boulevard, Richland, WA 99352, United States;Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2011

Quantified Score

Hi-index 31.47

Visualization

Abstract

We develop a new hierarchical reconstruction (HR) method [17,28] for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.