Numerical Methods For Stiff Singular Delay Differential Equations

This thesis includes five chapters. We have mainly studied numerical methods for stiff singular delay differential equations. A class of two-step continuity Runge-Kutta methods for solving singular and nonsingular delay differential equations, a class of two-step continuity Rosenbrock methods for solving stiff singular and nonsingular delay differential equations and a class of combined two-step continuity RK-Rosenbrock methods for solving stiff singular and nonsingular delay differential equations are presented. We choose general linear methods as integration formulae, continuous extension as dense-output. To maintain explicit computing procedure in the singular case, method of relaxing the effect of delay when computing the stages is presented. We use theory of trees, B-series, P-trees and P-series to deduce the order conditions of the methods constructed.In chapter one, the application and classification of delay differential equations are summarized. The actualities and development of numerical methods for solving singular delay differential equations and numerical methods for solving nonsingular delay differential equations are reviewed. The difficulties in solving delay differential equations are stated and numerical methods for solving singular delay differential equations are analyzed in emphasis. We put forward content will be investigated in the thesis in chapter one.In chapter two, a class of two-step continuity Runge-Kutta methods for solving singular and nonsingular delay differential equations is constructed. The methods constructed possess better numerical stability than continuous Runge-Kutta methods of same order when solving nonstiff delay differential equations and the cost of computing is same. The methods constructed maintain the property of explicit computation and avoid iteration when solving singular delay differential equations.In chapter three, a class of two-step continuity Rosenbrock methods for solving stiff singular and stiff nonsingular delay differential equations is constructed. The methods constructed are GP-stable when solving stiff nonsingular delay differential equations and do not need new interpolation procedures. Methods which are absolute stable in the current step are constructed, and combined with GP-stable methods, we presented numerical algorithm for solving stiff singular delay differential equations.In chapter four, a class of combined two-step continuity RK-Rosenbrock methods is constructed for solving stiff singular and stiff nonsingular delay differential equations respectively toward a partitioned system of stiff delay differential equations. Two-stepRunge-Kutta methods are used for solving nonstiff delay subsystem and two-step Rosenbrock methods are used for solving stiff delay subsystem.In chapter five, the conclusion and further research are presented.As regards the numerical methods presented above, we have made a systematic study on the construction of the methods, the specific formulas and their convergence and numerical stability. Numerical experiments conducted show that the methods we constructed are efficient.