Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

In this paper, we first introduce the problem class K p μ , λ , ζ with respect to the initial value problems of nonlinear impulsive differential equations in Banach spaces. The stability and asymptotic stability results of the analytic solution of the problem class K p μ , λ , ζ are obtained. Then, the numerical stability and asymptotic stability conditions of multistage one-step multiderivative methods are also given. Two numerical experiments are given to confirm the theoretical results in the end.

The numerical integration of stiff systems using stable multistep multiderivative methods

In this paper we describe the construction of stable multistep multiderivative methods designed for continuous numerical integration of stiff systems of initial value problems in ordinary dierential equations. These methods are obtained based on the multistep collocation technique, which are shown to be A-stable, convergent with large regions of absolute stability. They are suitable for solving stiff systems of initial value problems with large eigenvalues lying close to the imaginary axis. Numerical experiments illustrate the behaviour of the methods, which show that they are competitive with stiff integrators that are known to have strong stability characteristic properties. Comparison of the solution curves obtained is in good agreement with the exact solutions which demonstrate the reliability and usefulness of the methods.