Generalized Algebraic Kernels and Multipole Expansions for Massively Parallel Vortex Particle Methods

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 N-body problem. By the means of fast, multipole-based summation techniques, the unfavorable yet intrinsic O(N 2 )-complexity of these problems can be reduced to at least O(N logN). 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 O(N logN)-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 Julich 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.