Unbounded Rational Spectral Method

Recently more and more attentions are paid to problems in unbounded domain. In the context of spectral method, many numerical simulations have been provided and investigated. A simple way for their numerical simulations is to restrict calculations to some bounded subdomains and impose certain conditions on artificial boundaries. But this may destroy the accuracy for long time calculations. To remedy this deficiency, some authors used spectral methods accociated with orthogonal polynomial systems in unbounded domains, such as the Hermite and the Laguerre polynomials. It is also possible to reform such problems on infinite intervals to singular problems on finite intervals, and then used the Jacobi approximation to resolve them numerically.The main purpose of this work is to investigate the spectral method and pseudospectral method in unbounded domains by using some new mutually orthogonal systems of rational functions. We also give a framework for theoretical analysis of rational approximation.Firstly, we establish various weighted rational orthogonal systems. As some examples, we apply them to some linear model problems and some nonlinear problems such as the Burgers equation and the Klein-Gordon equation. The stability and the conver-gence of the proposed rational spectral and pseudospectral schemes are proved. The numerical results coincide very well with the theoretical analysis.Secondly, to solve some important nonlinear wave equations numerically, we establish various rational orthogonal systems with the Legendre weight. We also give some conservation and convergence analysis for the Dirac equation and the Korteweg-de Vries equation. The numerical results coincide very well with the theoretical analysis and demonstrate the efficiency of these new approaches.The main ideas and techniques used in this dissertation are also useful for other problems in undounded domains.