Numerical Stability And Convergence Analysis Of Symmetric Methods For Delay Differential Equations

Delay differential equations have been widely used in the fields of physics,biology,medicine,engineering and economy,e.t.c.Because of the complexity of the equation,it is difficult to obtain the analytical solution,so we have to use the numerical methods to solve the problem.The stability analysis of numerical methods is an important part.Since symmetric methods contain some good properties,it makes the analysis more standard and convenient.The dissertation is concerned with the delay-dependent stability of symmetric methods for several delay differential equations and the convergence of block boundary value methods for the index one neutral delay differential algebraic equations.The main contents of the dissertation include the following five aspects:Firstly,based on the linear delay integro-differential equations with real coefficients,the delay-dependent stability of symmetric boundary value methods is investigated.The symmetric boundary value methods are applied to solve the equation,including the Extended Trapezoidal Rules of first kind and second kind,the Top Order Methods and the B-spline linear multistep methods.Taking advantage of the boundary locus technique,the delay-dependent stability region of symmetric boundary value methods is obtained.It is proved that under suitable conditions,all the symmetric boundary value methods preserve the delay-dependent stability of the equation.Secondly,based on the linear neutral delay integro-differential equations with real coefficients,the delay-dependent stability of symmetric boundary value methods is investigated.By analyzing the properties of the boundary curves,an exact characterization of the stability region of symmetric boundary value methods is given.It is proved that under suitable conditions,the numerical methods can preserve the delay-dependent stability of the problem well.Furthermore,based on the linear neutral delay integro-differential equations with real coefficients,the delay-dependent stability of symmetric Runge-Kutta methods is investigated.The symmetric Runge-Kutta methods are applied to solve the equation,including the Gauss methods,the Lobatto IIIA?IIIB and IIIS methods.By using W-transformation and order star,an exact characterization of the delay-dependent stability region of high order symmetric Runge-Kutta methods is given.By comparing the relation between the analytical stability region and the numerical stability region,it is proved that the symmetric Runge-Kutta methods unconditionally preserve the delay-dependent stability of the problem.Next,based on a special kind of second order delay differential equations with three parameters,the delay-dependent stability of symmetric Runge-Kutta methods is investigated.The symmetric Runge-Kutta methods are applied to discrete the equation,and the delay-dependent stability region of the numerical methods is obtained.It is proved that high order symmetric Runge-Kutta methods whose stability function is a diagonal Pad?eapproximation unconditionally preserve the delay-dependent stability of the equation.Finally,we construct the block boundary value methods to solve the index one neutral delay differential algebraic equations.We extend the convergence analysis of Generalized Backward Differentiation Formulae to the case of general block boundary value methods.The error estimation of general block boundary value methods is obtained.