| Many phenomena in nature and engineering can be described by differential equations.For most differential equations,the exact solutions can not be given analytically,thus the study of the numerical methods for differential equations naturally becomes an important topic.Numerical integration method is a kind of numerical methods which constructed based on the variation of constants formula or equivalent integration equations.For example,the exponential integrators for the first-order semi-linear ordinary differential equations,the extended Runge-Kutta-Nystr?m methods for the second-order oscillatory differential equations and the product integration methods for the nonlinear fractional ordinary differential equations.In general,numerical integration methods have high accuracy,good stability and structure conservative properties.In this dissertation,several differential equations are solved numerically.A series of high order numerical integration methods are constructed.The convergence,stability and structure conservative properties of these algorithms are analyzed.The main contents are as follows:The convergence and stability of exponential general linear methods for delay differential equations are studied.Under some assumptions,it is proved that the convergence order and stage order of these methods for delay differential equations are equal to those for ordinary differential equations.For the linear test equations,the linear stability of exponential general linear methods are analyzed,and a sufficient condition is given.For the nonlinear delay differential equations,the GRN-stability of exponential general linear methods is proved.The multiderivative extended Runge-Kutta-Nystr?m methods for the second-order oscillatory differential equations are proposed.The methods are constructed based on the variation of constants formula,and make good use of the special structure brought by the linear term.These methods involve not only the right functions but also the derivatives of them.The adding derivatives make the methods obtain higher stage order and thus easy to achieve higher error order.The convergence,energy conservative property,stability and phase property of the methods are analyzed.A high order energy preserving scheme for nonlinear Riesz space fractional wave equations is constructed.A fourth order approximation for Riesz space fractional derivatives is proposed by using the weighted and shifted Lubich difference operators.By using this approximation,the nonlinear Riesz space fractional wave equations are semidiscreted.The stability and convergence of the semi-discrete system are investigated,and it is proved that the semi-discrete system preserves the semi-discrete energy exactly.Then,the fully-discrete scheme is obtained by using the extended Runge-Kutta-Nystr?m methods,the high order convergence and energy conservative property of the fully-discrete scheme are illustrated.The shifted Legendre product integration methods for nonlinear fractional ordinary differential equations are developed.The method is designed by considering the equivalent weakly singular Volterra integral equation and using the idea of local Fourier expansion.It is proved that the method achieves arbitrary high order in theory for smooth problems,and the good stability of the method is also proved. |
