Algorithms for scientific computing

Abstract

There has long been interest in algorithms for simulating physical systems. We are concernedwith two areaswithin this field: fastmultipolemethods andmeshlessmethods. Since Greengard and Rokhlin’s seminal paper in 1987, considerable interest has arisen in fast multipole methods for finding the energy of particle systems in two and three dimensions, and more recently in many other applications where fast matrix-vector multiplication is called for. We develop a new fast multipole method that allows the calculation of the energy of a system of N particles in O(N) time, where the particles’ interactions are governed by the 2D Yukawa potential which takes the form of a modified Bessel function Kv. We then turn our attention to meshless methods. We formulate and test a new radial basis function finite differencemethod for solving an eigenvalue problemon a periodic domain. We then applymeshlessmethods to modelling photonic crystals. After an initial background study of the field, we detail the Maxwell equations, which govern the interaction of the light with the photonic crystal, and show how photonic band gaps may be given rise to. We present a novel meshless weak-strong form method with reduced computational cost compared to the existing meshless weak form method. Furthermore, we develop a new radial basis function finite differencemethod for photonic band gap calculations. Throughout the work we demonstrate the application of cutting-edge technologies such as cloud computing to the development and verification of algorithms for physical simulations.

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