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Extended series analysis of full octahedral flow: Numerical evidence for hydrodynamic blowup. (English) Zbl 1032.76656

Summary: The power series in time of a full-octahedral flow, a candidate for finite-time blowup of the incompressible Euler equations, is analyzed. Sixty-five Taylor coefficients are found with a precision of 512 bits per Fourier coefficient (154 digits). This work is an extension of Pelz and Gulak (Phys. Rev. Lett. 79, 1998 (1997)), where 32 terms were found with half the precision. The solution is found in the form of a power series in time, Fourier series in space with the equation of motion written as a quadratic operation on the Fourier–Taylor coefficients of a single component of vorticity. Particular attention to precision and roundoff is given. In agreement with previous work, Padé approximants of enstrophy series consistently show an isolated, simple pole is located at a real time of about two. Similar findings exist for the series representation of vorticity derivatives at the origin, the purported blowup point. Despite the lack of convergence proofs for Padé resummation, consistent evidence typically yields a valid continuation. Higher-order Sobolev norms of vorticity, however, do not show a similar behavior. The results are also difficult to explain in terms of similarity scalings.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76B47 Vortex flows for incompressible inviscid fluids
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