Highly Efficient Numerical Methods For Stiff Differential Equations And Analysis Of Numerical Stabilities For Neutral Functional Differential Equations

Stiff differential equations can be found in the high-tech fields such as aviation, spaceflight, thermonuclear reaction, automatic control, electronic network and chemical kinetics and so on. The numerical methods for these stiff problems are undoubtedly important. The large-scale ordinary differential equations derived from the semi-discretization of partial differential equations are another important sources of stiff differential equations. In the last few decades, a lots of important results on the theory of computational methods for stiff differential equations have been obtained. The absolute stability regions are required as big as possible for computing the stiff problems and the numerical methods are required to be extremely stable at∞. Therefore the construction of highly efficient numerical methods which have advantages on both sides is always one of important research subject of stiff problems.Some famous numerical methods such as BDF and etc are extremely stable at∞. But the stability regions of high-order methods are not ideal enough. Although Runge-Kutta methods of Gauss type are A-stable, they are not strongly stable at∞. So the construction of highly efficient algorithm which has strong stability at∞and bigger stability region is the first work of this dissertation:(1) Two classes of improved backward differentiation formulae are presented, whose abbreviation are IBDF1 and IBDF2 respectively;(2) The improved Enright methods with order 5-9 are presented;The above new methods preserve the original advantages of BDF and Enright methods respectively and their stability regions have been achieved some large improvements. So the application prospect is very extensive.(3) Several classes of multistep Runge-Kutta methods of Gauss type which have the optimal stability at∞are obtained by using pattern search method. The stability at∞of these new methods is superior to the one-step Runge-Kutta methods of Gauss type. Theoretical analysis and numerical experiments show that the former's actual calculation accuracy are far higher than the latter for computing strongly stiff problems.It is should noticed that there does not exist the multistep Runge-Kutta method of Gauss type which are L-stable at∞. It is beneficial to obtain these strongly stable methods which stability matrix's spectral radius is minimal at∞. The Neutral Functional Differential Equations (NFDEs) often arise in biology, physics, control theory, engineering technology and so on. In the last few years, many authors have investigated the linear stability of numerical methods and obtained a lot of important results. Recently, some authors have studied the numerical stability of nonlinear neutral delay differential equations (NDDEs) and neutral delay integro-differential equations (NDIDEs). The further research of numerical stability for NDIDEs and more general NFDEs on basis of these results is an another work of this dissertation:(4) The numerical stability results of linear multistep methods for NDIDEs are obtained. Theoretical analysis shows that A-stable linear multistep methods are also asymptotically stable if the problems are asymptotically stable.(5) The nonlinear stability of explicit and diagonally implicit Runge-Kutta methods for neutral functional differential equations (NFDEs) are discussed in Banach spaces. The results on the numerical stability and conditional contractibility of some explicit and diagonally implicit Runge-Kutta methods for nonlinear NFDEs are obtained. Numerical examples are given to confirm the theoretical results.It should be pointed out that the other related literatures of home and abroad mainly discussed the stability of numerical methods for NDDEs. The research of numerical stabilities of general nonlinear NFDEs in Banach space haven't been seen yet.