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Uniform high-order spectral methods for one- and two-dimensional Euler equations. (English) Zbl 0766.76069

Summary: We study uniform high-order spectral methods to solve multidimensional Euler gas dynamics equations. Uniform high-order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the essentially non-oscillatory polynomial interpolations into the spectral methods. Based on the new approximations, we propose nonoscillatory spectral methods which possess the properties of both upwinding difference schemes and spectral methods. We present numerical results for inviscid Burgers’ equation, various one- dimensional Euler equations including the interactions between a shock wave and density disturbances, Sod’s and Lax’s and blast wave problems. Finally, we simulate the interaction between a Mach-3 two-dimensional shock wave and a rotating vortex.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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