Convergence And Stability Of The IMEX One-leg Methods For Stiff Volterra Delay Integro-differential Equations

Integral differential equations(IDEs) are an important branch of modern mathematics. The researches into theory, computation and applications of IDEs have been valued. Delay integro-differential equations(DIDEs) have appeared and been applied to some science and engineering fields such as biology, dynam-cs,automatic control, electric system, physics and spatial discretization of initial oundary value problems of partial functional differential equations. Some of DIDEs are of stiffness,and their right functions can be split to stiff and non stiff parts. In order to make the computation more efficient, we often adopt implicit-explicit methods, i.e. the stiff part is integrated implicitly and the non stiff part s integrated explicitly.Now, the commonly used implicit-explicit methods mainly include two class-es:the implicit-explicit linear multistep methods and the implicit-explicit Runge-Kutta methods. In this paper, our interest pours into studying the convergence and stability of the implicit-explicit one-leg methods for a class of initial value problems of stiff Volterra delay integro-differential equations. These extend the corresponding results in the existing literature. The full text is composed of four chapters.In Chapter 1, we introduce the current research situation and the main work of this paper.In Chapter 2, we introduce a class of initial value problems of stiff Volterra delay integro-differential equations and the corresponding implicit-explicit one-leg methods.In Chapter 3, the errors and stability of the implicit-explicit one-leg methods are analyzed. Then we prove that the implicit-explicit one-leg methods are stable and convergent of order 2.In Chapter 4, we give some numerical experiments to confirm the obtained stability and convergence results.