Lagendre-Gauss And Generalized Laguerre-Gauss Collocation Methods For IVPs Of Second-order ODEs

The numerical methods for solving the differential equations is one of the subjects of computational mathematics,as well as an important part of large-scale scientific computing.In this thesis,the spectral method,which is one of the three main numerical methods for differential equations,is investigated.The fascinating merit of spectral method is its high accuracy,i.e.,the smoother the exact solution,the more accurate the numerical results.Therefore,it has been applied successfully to numerical simulations of many practial problems,such as the numerical simulation of quantum mechanics, weather prediction,ocean problems,and so on.Many practical problems arising in science and engineering are governed by initial value problems(IVPs) of ordinary differential equations(ODEs).There have been fruitful results on numerical methods for first order ODEs,see,e.g.,Butcher[6-8], Hairer,Norsett and Wanner[46],Hairer and Wanner[47],Higham[49],Lambert[57], Stetter[80]and Stuart and Humphries[82].For Hamiltonian systems,we refer to the symplectic difference method,see Feng[26],Feng and Qin[25],Hairer,Lubich and Wanner[45],and Sanz-Serna and Calvo[71].It is also important and interesting to consider IVPs of second order ODEs.For instance,after spacial discretization,many nonlinear wave equations,such as Klein-Gorden and sine-Gorden equations,are reduced to certain systems of second order ODEs.We may reformulate such problems to some systems of first order ODEs and then solve them numerically.Whereas,for saving work,it seems reasonable to solve them directly sometimes,see,e.g.,Fehlberg[22],Kramarz[55],Kuo and Vazquez[56], Franco[26],Vigo-Aguiar and Ramos[84],and Konguetof and Simos[54].However,so far,there is no numerical method with the spectral accuracy,for initial value problems of second order ODEs.The main difficulty is how to design proper algorithms and analyze their numerical errors precisely.In this thesis,we proposed two numerical integration algorithms for the IVPs of second-order ODEs.Numerical simulations indicate their reliable efficiency,and coincide well with theoretical analysis.In the first chapter,we outline the motivation and the main results.In the second chapter,we study a new numerical method for IVPs of second-order ODEs.We proposed a new collocation method based on Legendre-Gauss interpolation, and proved its spectral accuracy.Numerical results indicate the merits of this approach, in other words,it saves a lot of computational time and provides more accurate results than many commonly used algorithms.In the third chapter,we develop a new collocation methods based on the generalized Laguerre-Gauss interpolation approximation for IVPs of second-order ODEs.Numerical results illustrate its spectral accuracy.We also take the generalized Laguerre functions as basis functions to design another collocation method.Numerical examples also demonstrate its effectiveness.