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

doi:10.1016/S0252-9602(10)60014-8

Abstract

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

Zhiliang Xu, Guang Lin. SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS[J].Acta mathematica scientia,Series B, 2009, 29(6): 1737-1748.

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