Spectral Methods In Unbounded Domains

Many problems in science and engineering are set in unbounded domains. The simplest method to deal with them is to set some artificial boundaries, impose certain artificial boundary conditions and then resolve them numerically. Whereas these treatments may cause additional errors. The main purpose of this work is to develop the spectral methods associated with some orthogonal systems of polynomials in unbounded domains. We start by the Hermite polynomials interpolation approximation. As an example, we apply it to the Burgers equation on the whole line. The stability and the spectral accuracy of the proposed scheme are proved. The numerical results show the high accuracy of this approach. We derive the Laguerre interpolation approximation. The pseudospectral scheme for the BBM equation on the half line is discussed. We also give a theoretical result for the stream function form of the Navier-Stokes equation in unbounded domains. The existence, uniqueness and the regularility are studied. Its mixed Laguerre-Legendre spectral scheme and mixed Laguerre-Legendre pseudospectral scheme are also constructed. We prove the stability and convergence of the proposed schemes. The numerical results show the efficiency of these approach. The main idea and techniques used in this work are also applicable to other nonlinear partial differential equations in unbounded domains.