Legendre Spectral Configuration Method For Initial Value Problems Of Second-Order Nonlinear Ordinary Differential Equations And Its Application In Wave Equations

As an important mathematical model,the initial value problem(IVPs)of second-order nonlinear ordinary differential equations(ODEs)comes from celestial mechanics,quantum mechanics,electronics,ecology and other disciplines,and it has a very wide range of applications in many fields of science and engineering computation.In addition,after spacial discretization,a large class of second-order wave equations are reduced to certain systems of second-order ODEs.Therefore,it is of great theoretical and practical significance to study the numerical methods of the IVPs of second-order ODEs.In this thesis,we shall mainly study the Legendre spectral collocation method for IVPs of second-order nonlinear ODEs and its application to nonlinear wave equations as follows.On the one hand,we first propose a single-interval spectral collocation scheme based on the Legendre-Gauss-Radau points for a class of IVPs of second-order nonlinear ODEs,design a fast and efficient iterative algorithm by using the Legendre expansions,and we further design a multi-interval spectral collocation scheme.Secondly,we carry out a rigorous error analysis of the single-interval spectral collocation method.Finally,we demonstrate the high-order accuracy of the Legendre spectral collocation method through a series of numerical examples.In particular,the numerical results show that the multi-interval Legendre spectral collocation method can deal with problems with oscillatory solutions,singular solutions,as well as numerical simulations of long time behaviors.On the other hand,we consider a class of second-order nonlinear evolution equations.We use the Legendre-Gauss-Radau spectral collocation method proposed above for temporal discretization and use the Legendre-Gauss-lobatto spectral collocation method for spatial discretization,which leads to a fully discrete spectral collocation approximation.We then take two typical nonlinear wave equations,such as Sine-Gordon equation and Klein-Gordon equation as examples,the numerical results show that the proposed space-time spectral collocation method has high-order accuracy in both time and space,and it is stable for long time numerical simulation.