RESEARCH, IMPLEMENTATION AND ASSESSMENT OF A FAST SUMMATION METHOD APPLIED TO THE VORTEX BLOB METHOD

Abstract

A large set of problems of great interest to the scientific computing com<U+00AD>munity involve calculations that are usually too great, and the computational resources too small, to complete them in a sensible time frame. This is the motivation behind the development of clever algorithms that make an effi<U+00AD>cient use of the available resources in order to find a solution in a shorter time.This research will have its focus on the research, implementation and as<U+00AD>sessment of a clever algorithm that can speed up the calculations performed by the Vortex Blob Method (VBM). The problem involves the so<U+00AD>lution of an N-body problem, where a naive implementation will perform O(N2) operations. Where in the case of the VBM, usual experiments re<U+00AD>quires values of N in the order of thousands or millions, which makes the naive approach unfeasible.In order to solve problems for large N values within reasonable times it is necessary to utilize a computational method which requires fewer than O (N2) operations. This work will explore the use of the Fast Multipole Method (FMM) -a highly accurate O (N) algorithm- to speed up the calculation performed in the computation of the velocity field in the VBM. The FMM is considered one of the top ten most significant algorithms in scientific computation of the 20th century. The FMM and their applications are an active area of research. Key Words: N -body problems, Fast Summation Methods, Fast Multipole Method, Vortex Blob Method, Scientific Computing, Computational Fluid Dynamics.