A Multiple Interval Chebyshev-gauss-lobatto Collocation Method For Ordinary Differential Equations

Spectral methods possess high order accuracy, and have become increasingly popular inspatial discretization of differential equations. For time-dependent partial differential equations(PDEs), one usually uses spectral methods to approximate the solutions in space and finite differ-ence approaches to march the solutions in time. This tactic results in an unbalanced scheme: it hasinfinite accuracy in space and finite accuracy in time.To overcome this disadvantage, some authors developed spectral methods for time-discretization of time-dependent PDEs, see [1,6,8,16,19,21–26,36]. Moreover, some highorder numerical methods for the initial value problems of ordinary differential equations (ODEs)are also established, see [10–14,17,27,30,31,33,34].The aim of this dissertation is to propose a multiple interval Chebyshev-Gauss-Lobatto spec-tral collocation method for the initial value problems of ODEs. We propose an efficient algorithmand present the error estimate for the hp-version of the multiple interval collocation method. Nu-merical results exhibit that the scheme is stable for long-time calculations and possesses high orderaccuracy. Moreover, it is also particularly attractive for ODEs with highly oscillating solutions,steep gradient solutions and nonsmooth solutions.This dissertation is organized as follows. In the next chapter, we propose the multiple intervalChebyshev-Gauss-Lobatto spectral collocation scheme, and present some approximation results onthe Chebyshev-Gauss-Lobatto interpolation. The convergence analysis for the suggested methodis given in Chapter3. Numerical experiments are carried out in Chapter4, which confirm thetheoretical expectations. The final chapter is for some concluding remarks.