On the Rokhlin-Greengard method with vortex blobs for problems posed in all space or periodic in one direction (Q1902678)

The authors consider Rokhlin-Greencard (\(R\)-\(G\)) fast multipole algorithm to evaluate vortex blob interactions in a two-dimensional incompressible inviscid fluid. The \(R\)-\(G\) method requires \(O (n)\) floating point operations to evaluate the velocity field induced by the interaction of \(n\) blobs, whereas the direct evaluation procedure requires \(O (n^2)\) floating point operations. An exact solution of the incompressible Euler equation is used to demonstrate that the \(R\)-\(G\) algorithm computes vortex blob interactions accurately. But the structure of the vortex blobs forces a practical limitation on the highest bisection level in the \(R\)-\(G\) algorithm, a restriction which does not apply when point vortices are used. If this restriction is violated, then the errors arising from the differences between the \(R\)-\(G\) evaluation and that of the direct method can be of the same order of magnitude as the errors arising from the vortex blob method itself, or even larger. The \(R\)-\(G\) algorithm with blobs is also extended to two-dimensional problems which are periodic in one direction and unbounded in the other. The computational details and performance of the algorithm are presented in detail.