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Non-oscillatory spectral element Chebyshev method for shock wave calculations. (English) Zbl 0781.65083
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 W. Cai, D. Gottlieb and 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.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35L67 Shocks and singularities for hyperbolic equations
76L05 Shock waves and blast waves in fluid mechanics
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