Chebyshev-Gauss Collocation Method For Ordinary Differential Equations

The spectral method is one of the main numerical methods for partial differential equations.Its main merit is the spectral accuracy. In the past few decades, spectral method has become in-creasingly popular and been widely used in the numerical simulations of fluid dynamics, quantummechanics, mathematical finance and other issues.Among the existing methods, numerical schemes based on Taylor’s expansions or quadratureformulas have been frequently used. However, so far, there is very few method with spectralaccuracy, for solving initial value problems of ordinary differential equations.Recently, Guo Ben-yu et al. developed several Laguerre/Legendre spectral collocation meth-ods for ordinary differential equations based on Laguerre/Legendre expansions, which are usuallymore stable and effective than the classical implicit Runge-Kutta methods.Motivated by the precious spectral collocation method, we propose in this dissertation thesingle-step and multiple-domain Chebyshev-Gauss spectral collocation method for ordinary dif-ferential equations. We also suggest several new algorithms, which can be implemented in a stableand efficient manner. Numerical illustrations also show that the suggested algorithms are particu-larly attractive for ODEs with stiff and/or long-time behaviors.This work consists of four partsIn Chapter Ⅰ, we recall some classical methods for ordinary differential equations.In Chapter Ⅱ, we construct a single-step Chebyshev-Gauss collocation method for ordinarydifferential equations. We propose three new algorithms, and analyze the numerical error. We alsoprovide some numerical results to justify our theoretical analysis.In Chapter Ⅲ, we describe a multi-domain Chebyshev-Gauss collocation scheme for ordinarydifferential equations. Numerical experiments with stiff and long time behaviors show that ournew algorithms are more efficient and accurate over some popular existing methods.In Chapter Ⅳ, we present some concluding discussions.