Spectral Collocation Methods Based On Birkhoff Interpolation For Two Classes Of Wave Equations

Wave equations are widely used for describing various phenomena of wave propagation.In this dissertation,we study two typical wave equations,including the Schr?dinger equation and the Korteweg-de Vries equation.They arise in many fields,including quantum mechanics,nonlinear optics,plasma physics,superconductivity and crystals.Spectral collocation method is one of widely used numerical methods because of its exponential convergence.However,the coefficient matrices produced by the traditional polynomial based spectral collocation methods are full and ill-conditioned.In this dissertation,we construct a set of non-polynomial basis functions based on Birkhoff interpolation.Using these basis functions we build a new spectral collocation method.The new collocation scheme leads to well-conditioned linear systems for general boundary value problems.Moreover,various boundary conditions are satisfied exactly.On the other hand,in many cases,the physical domain is unbounded.When considering numerical methods for wave equations set on unbounded domain,it is required to truncate the whole space into finite domain because of the limited computing resources.Then it is desired to supply some artificial boundary conditions to the bounded domain.If inappropriate boundary conditions are imposed on boundary,the outgoing waves will be reflected back into the computational domain,which is not consistent with the underlying equation initially defined on unbounded domain.One has to minimize these spurious reflection waves as much as possible.If the imposed boundary conditions are such that there is no spurious reflections arise at the boundary,one refers to these boundary conditions as transparent boundary conditions.For the nonlinear Schr?dinger equation defined on finite interval,based on the secondorder generalised Birkhoff interpolation,a set of non-polynomial basis functions are constructed.To study the interpolation error for these basis functions,the stability and convergence analysis in non-uniformly weighted Sobolev space are preformed for the JacobiGauss-Lobatto interpolation operator.With the help of these analysis,the error estimates for the non-polynomial basis functions are built in the weighted Sobolev space.For the linear Schr?dinger equation proposed on one-dimensional unbounded interval,after discretization in time,the derivation of the time semi-discrete transparent boundary conditions are presented.Under these boundary conditions,the stability of the time semidiscrete system are proved.Finally,the new spectral collocation method using nonpolynomial basis functions are adopted for spatial discretization.That results in the fully discrete scheme satisfying the boundary conditions exactly.Moreover,the fully discrete scheme collapses to an explicit scheme by choosing the free parameter in the basis functions properly.Numerical tests are carried out to demonstrate the high accuracy and transparency for the reflection waves of the proposed methods.For the linear Korteweg-de Vries equation on one-dimensional unbounded interval,after temporal discretization,the original unbounded problem is converted to equivalent problem consisting in interior subproblem and two exterior subproblems.The desired transparent boundary conditions for the interior subproblem are obtained by solving two exterior subproblems through the Z transformation.The stability of the time semi-discrete system with transparent boundary conditions are built.In the Korteweg-de Vries equation case,there is no explicit expression for the inverse Z transformation in the transparent boundary conditions.A stable and accurate algorithm to compute the inverse Z transformation is given.Then we consider the spatial discretization of the time semi-discrete scheme for the Korteweg-de Vries equation.Based on the third-order generalised Birkhoff interpolation,another set of non-polynomial basis functions are constructed.The interpolation error estimates in non-uniformly weighted Sobolev space for these basis functions are built.Equipped with the new non-polynomial basis functions,the fully discrete spectral collocation scheme for the Korteweg-de Vries equation are constructed.The fully discrete scheme preserves the transparency for the reflection waves.At the same time,the basis functions build in two free parameters intrinsically which can be chosen properly so that the fully discrete scheme collapses to an explicit scheme.Numerical results are presented to demonstrate the effectiveness and high accuracy of the proposed method.