$B$-theory of numerical methods for stiff Volterra functional differential equations

Shoufu Li

Department of Mathematics, Xiangtan University, Xiangtan 411105, P.R. China

lisf@xtu.edu.cn

Contributed talk

$B$-theory of numerical methods for stiff Volterra functional differential equations are established which provide unified theoretical foundation for the error and stability analysis of numerical methods when applied to nonlinear stiff initial value problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and functional differential equations(FDEs) of any other type which appear in practice.

The main results are distinguished in three parts:

1.  $B$-theory of Runge-Kutta methods for FDEs,

2.  $B$-theory of general linear methods for FDEs,

3.  Asymptotic stability of algebraically stable methods for FDEs,

and will be published in three papers respectively.

The first two parts can be regarded as extension of $B$-theory of numerical methods for ODEs presented by Frank et al. in 1984 and by the author in 1988, and the main theoretical framework can be described by the following implications:

$\quad \,\,{\rm diagonal~ stability}\Rightarrow \left\{ \begin{array}{l} BSI{\rm -stability}\\ BS{\rm -stability} \end{array}\right.$

$\left. \begin{array}{l} \left. \begin{array}{l} {\rm algebraic~ stability~~~}... ...istency} \end{array}\right\}\Rightarrow p{\rm -th~ order}~ B{\rm -convergence}$

So far we have never seen any similar study for stiff FDEs and IDEs, but there have been a lot of similar work on stiff DDEs in literature. However, the existing results for nonlinear stiff DDEs are usually more restricted. For example, the stability results of Runge-Kutta methods for DDEs obtained by Chengming Huang et al. appeared in BIT 1999, only apply to DDEs with a constant delay and a negative one-sided Lipschitz constant, and only apply to Runge-Kutta methods with a linear interpolation operator and a fixed stepsize. In comparison with the results obtained directly from the general $B$-theory for FDEs established in the present paper, we find that the latter have broken all the aforementioned restrictions so that they are more extensive and deeper.