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Tracking discontinuities in hyperbolic conservation laws with spectral accuracy. (English) Zbl 1343.76039

Summary: It is well known that the spectral solutions of conservation laws have the attractive distinguishing property of infinite-order convergence (also called spectral accuracy) when they are smooth (e.g., [C. Canuto et al., Spectral methods in fluid dynamics. New York etc.: Springer-Verlag (1988; Zbl 0658.76001); J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., New York: Dover (2001; Zbl 0994.65128); C. Canuto et al., Spectral methods: fundamentals in single domains. Berlin: Springer (2006; Zbl 1093.76002)]). If a discontinuity or a shock is present in the solution, this advantage is lost. There have been attempts to recover exponential convergence in such cases with rather limited success. The aim of this paper is to propose a discontinuous spectral element method coupled with a level set procedure, which tracks discontinuities in the solution of nonlinear hyperbolic conservation laws with spectral convergence in space. Spectral convergence is demonstrated in the case of the inviscid Burgers equation and the one-dimensional Euler equations.

MSC:

76M22 Spectral methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
76L05 Shock waves and blast waves in fluid mechanics
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