中文摘要：

一类重要的常微分方程源自用线方法求解非线性双曲型偏微分方程，这类常微分方程的解具有单调性，因此要求数值方法能保持原系统的这种性质．本文研究多步Runge—Kutta方法求解常微分方程初值问题的保单调性．分别获得了多步Runge—Kutta方法是条件单调和无条件单调的充分条件．

英文摘要：

An important class of ordinary system is that whose solutions satisfy a monotonicity property for a given norm. The system arises from the discretization of the spatial derivatives in the hyperbolic partial differential equations. For these problems, a natural requirement for the numerical solution is the reflection of this monotonicity property, perhaps under certain stepsize restriction. This paper deals with the monotonicity property of multistep Runge- Kutta methods. Sufficient conditions are given for multistep Runge-Kutta methods to be conditional monotonicity preserving and unconditional monotonicitv preserving,respectively.