Non-oscillatory spectral element Chebyshev method for shock wave calculations
A new algorithm on a non-oscillatory spectral element discretization is developed for the solution of hyperbolic partial differential equations. A conservative formulation is proposed based on cell averaging and reconstruction procedures, that employ a staggered grid of Gauss- Chebyshev and Gauss-Lobatto-Chebyshev discretizations. The non-oscillatory reconstruction procedure is based on ideas similar to those proposed by \textit{W. Cai, D. Gottlieb} and \textit{C.-W. Shu} [Math. Comput. 52, No. 186, 389-410 (1989; Zbl 0666.65067)] but employs a modified technique which is more robust and simpler in terms of determining the location and strength of a discontinuity. It is demonstrated by model problems of linear advection, the inviscid Burgers equation and a one-dimensional Euler system that the proposed algorithm leads to stable, non-oscillatory results. Exponential accuracy away from the discontinuity is realized for the inviscid Burgers' equation example.
non-oscillatory spectral element Chebyshev method
scientific article; zbMATH DE number 404846