Acta mathematica scientia,Series B ›› 2009, Vol. 29 ›› Issue (6): 1737-1748.doi: 10.1016/S0252-9602(10)60014-8

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SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

 Zhiliang Xu, Guang Lin   

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA; Computational Mathematics Group, Pacific Northwest National Laboratory, 902 Battelle Boulevard, Richland, WA |99352, USA
  • Received:2009-11-03 Online:2009-11-20 Published:2009-11-20
  • Supported by:

    Research was supported in part by NSF grant DMS-0800612.  Research was supported by Applied Mathematics program of the US DOE Office of Advanced Scientific Computing Research. The Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy under Contract DE-AC05-76RL01830.

Abstract:

The hierarchical reconstruction (HR) [Liu, Shu, Tadmor and Zhang, SINUM '07] has been successfully applied to prevent oscillations in
solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential non-oscillation  is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions  without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.

Key words: spectral/hp element method, hierarchical reconstruction, discontinuous Galerkin, hyperbolic conservation laws

CLC Number: 

  • 65M60
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