Buttke, Thomas F. A numerical study of superfluid turbulence in the self-induction approximation. (English) Zbl 0639.76136 J. Comput. Phys. 76, No. 2, 301-326 (1988). Two stable numerical methods are presented to solve the self-induction equations of vortex theory. These numerical methods are validated by comparison with known exact solutions. A new self-similar solution of the self-induction equation is presented and the approximate solutions are shown to converge to the exact solution for the self-similar solution. The numerical method is then generalized to solve the equations of motion of a superfluid vortex in the self-induction approximation where reconnection is allowed. A careful numerical study shows that the mesh spacing of the method must be restricted so that the approximate solutions are accurate. The line length density of a system of superfluid vortices is calculated. Contrary to earlier results it is found that the line length density produced does not scale as the velocity squared and therefore is not characteristic of homogeneous turbulence. It is concluded that the model equation used is inadequate to describe superfluid turbulence. Cited in 23 Documents MSC: 76Y05 Quantum hydrodynamics and relativistic hydrodynamics 81V99 Applications of quantum theory to specific physical systems 76M99 Basic methods in fluid mechanics 76F99 Turbulence Keywords:stable numerical methods; self-induction equations of vortex theory; exact solutions; self-similar solution; approximate solutions; superfluid vortex; self-induction approximation; superfluid turbulence × Cite Format Result Cite Review PDF Full Text: DOI Link References: [1] Feynman, R. P., Application of Quantum Mechanics to Liquid Helium, (Progress in Low Temperature Physics, Vol. 1 (1955), North-Holland: North-Holland Amsterdam) · Zbl 0058.44807 [2] Schwarz, K. W., Phys. Rev. B, 18, 245 (1978) [3] Schwarz, K. W., Phys. Rev. Lett., 49, 283 (1982) [4] Chorin, A. J.; Marsden, J., A Mathematical Introduction to Fluid Mechanics (1979), Springer-Verlag: Springer-Verlag New York · Zbl 0417.76002 [5] Hama, F. R., Phys. Fluids, 5, 1156 (1962) · Zbl 0149.45302 [6] Hama, F. R., Phys. Fluids, 6, 526 (1963) · Zbl 0116.42701 [7] Batchelor, G. K., An Introduction to Fluid Dynamics (1967), Cambridge Univ. Press: Cambridge Univ. Press London · Zbl 0152.44402 [8] Chorin, A. J., Commun. Math. Phys., 83, 517 (1982) · Zbl 0494.76024 [9] Siggia, E., Phys. Fluids, 28, 794 (1985) · Zbl 0596.76025 [10] Hasimoto, H., J. Fluid Mech., 51, 477 (1972) · Zbl 0237.76010 [11] Newell, A. C., Solitons in Mathematics and Physics (1985), SIAM: SIAM Philadelphia · Zbl 0565.35003 [12] Buttke, T. F., Thesis, (LBL Report LBL-22086 (1986), University of California: University of California Berkeley), (unpublished) [13] Hall, H. E.; Vinen, W. F., (Proc. R. Soc. A, 238 (1956)), 204 [14] Anderson, C.; Greengard, C., SIAM J. Numer. Anal., 22, 413 (1985) · Zbl 0578.65121 [15] Schwarz, K. W., Phys. Rev. B, 31, 5782 (1985) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.