常微分方程初值问题的一类单步解法

Class of single step schemes for initial value problems of ordinary differential equations

利用Taylor展开工具提出常微分方程初值问题的一类单步求解法,并证明该方法的相容性、稳定性和收敛性.该方法克服了传统单步法的缺点,既不使用高阶导数,同时在每一步计算时使用的函数值的个数又明显少于传统方法的个数,其绝对稳定区间均大于同阶的Adams外插法的绝对稳定区间.数值结果表明:该方法具有较高的精度.另外,此方法还可以推广到方程组和高阶的情形.

By Taylor expansion,this paper introduces a better class of single step schemes for solving the initial value problems of ordinary differential equations,and proves the consistency,stability and convergence of these methods.These schemes overcome the weakness of the traditional single step schemes：The new methods don′t require the partial derivatives of the function.The number of function values used in each step in the computation is also much less than the order number of the method,and the intervals of their absolute stability are larger than those of the Adams methods.Numerical results show that these methods have the better accuracy.These methods can be extended the situation of systems of equations.