An effective spectral collocation method for the direct solution of high-order ODEs

This paper reports a new Chebyshev spectral collocation method for directly solving high-order ordinary differential equations (ODEs). The construction of the Chebyshev approximations is based on integration rather than conventional differentiation. This use of integration allows the multiple boundary conditions to be incorporated more efficiently. Numerical results show that the proposed formulation significantly improves the conditioning of the system and yields more accurate results and faster convergence rates than conventional formulations

Discontinuous collocation methods and gravitational self-force applications

Numerical simulations of extereme mass ratio inspirals, the mostimportant sources for the LISA detector, face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a-priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.Comment: 29 pages, 5 figure

逐次探索による多目的最適化および宇宙往還機の複合領域概念設計への応用

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 土屋 武司, 東京大学教授 鈴木 真二, 東京大学教授 鈴木 宏二郎, 東京大学准教授 今村 太郎, 防衛大学校准教授 横山 信宏University of Tokyo(東京大学