Block Theta-methods And Numerical Analysis Of Differential-algebraic Equations

Differential-algebraic equations play an important role in the mathematical modeling of many scientific and engineering problems,and have a wide range of applications in many fields,such as multi-body mechanics,electrical design,chemical reaction systems,biology,and biomedicine,etc.However,in many problems,the systems we consider contain delay terms,usually described by delay differential-algebraic equations.Delay differential-algebraic equations can be regarded either as differential-algebraic equations with delays or as delay differential equations with constraints,so their structure is more complicated.In addition to linear multistep methods and Runge-Kutta methods,block methods are the third type of numerical methods for solving ordinary differential equations,and their advantages mainly include high precision,good stability and parallelism.This thesis is concerned with the convergence and numerical stability of the block ?-methods for solving differential-algebraic equations and delay differential-algebraic equations.In Chapter 1,we briefly introduce the basic theory of differential-algebraic equations and delay differential-algebraic equations and their numerical methods,and review some basic results on the block ?-methods for solving ordinary differential equations.In Chapter 2,we propose and obtain the convergence of the block ?-methods for the differential-algebraic equations of index 1 and index 2,respectively.Based on the linear constant coefficient differential-algebraic equations,we present the necessary and sufficient condition such that the block ?-methods are absolutely stable and conduct numerical examples to verify the theoretical results.In Chapter 3,in order to preserve the single-step nature of the block methods,we exploit the continuous extension of the block ?-methods to approximate the delay term to construct a class of block ?-methods for solving the delay differential-algebraic equations,and establish the convergence of the numerical schemes for solving the problems of index 1 and index 2.At the same time,based on linear constant coefficient delay differential-algebraic equations,we study the numerical stability of the block ?-methods and obtain the necessary and sufficient condition for the numerical schemes to be absolutely stable.Numerical results are given to confirm our theoretical results.