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@PHDTHESIS{Speck:16252,
      author       = {Speck, Robert},
      title        = {{G}eneralized {A}lgebraic {K}ernels and {M}ultipole
                      {E}xpansions for {M}assively {P}arallel {V}ortex {P}article
                      {M}ethods},
      volume       = {7},
      school       = {Universität Wuppertal},
      type         = {Dr. (Univ.)},
      address      = {Jülich},
      publisher    = {Forschungszentrum Jülich GmbH Zentralbibliothek, Verlag},
      reportid     = {PreJuSER-16252},
      isbn         = {978-3-89336-733-7},
      series       = {Schriften des Forschungszentrums Jülich. IAS Series},
      pages        = {IV, 115 S.},
      year         = {2011},
      note         = {Record converted from JUWEL: 18.07.2013; Universität
                      Wuppertal, Diss., 2011},
      abstract     = {Regularized vortex particle methods offer an appealing
                      alternative to common mesh-based numerical methods for
                      simulating vortex-driven fluid flows. While inherently
                      mesh-free and adaptive, a stable implementation using
                      particles for discretizing the vorticity field must provide
                      a scheme for treating the overlap condition, which is
                      required for convergent regularized vortex particle methods.
                      Moreover, the use of particles leads to an $\textit{N}$
                      -body problem. By the means of fast, multipole-based
                      summation techniques, the unfavorable yet intrinsic
                      $\mathcal{O}$($\textit{N}$ $^{2}$)-complexity of these
                      problems can be reduced to at least
                      $\mathcal{O}$($\textit{N}$ log $\textit{N}$). However, this
                      approach requires a thorough and challenging analysis of the
                      underlying regularized smoothing kernels. We introduce a
                      novel class of algebraic kernels, analyze its properties and
                      formulate a decomposition theorem, which radically
                      simplifies the theory of multipole expansions for this case.
                      This decomposition is of great help for the convergence
                      analysis of the multipole series and an in-depth error
                      estimation of the remainder. We use these results to
                      implement a massively parallel Barnes-Hut tree code with
                      $\mathcal{O}$($\textit{N}$ log $\textit{N}$)-complexity,
                      which can perform complex simulations with up to 10$^{8}$
                      particles routinely. A thorough investigation shows
                      excellent scalability up to 8192 cores on the IBM Blue
                      Gene/P system JUGENE at Jülich Supercomputing Centre. We
                      demonstrate the code’s capabilities along different
                      numerical examples, including the dynamics of two merging
                      vortex rings. In addition, we extend the tree code to
                      account for the overlap condition using the concept of
                      remeshing, thus providing a promising and mathematically
                      well-grounded alternative to standard mesh-based
                      algorithms.},
      cin          = {JSC},
      ddc          = {500},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {Scientific Computing (FUEK411) / 411 - Computational
                      Science and Mathematical Methods (POF2-411)},
      pid          = {G:(DE-Juel1)FUEK411 / G:(DE-HGF)POF2-411},
      typ          = {PUB:(DE-HGF)11 / PUB:(DE-HGF)3},
      url          = {https://juser.fz-juelich.de/record/16252},
}