基于勒让德多项式逼近的4级4阶隐式Runge-Kutta方法

A FOUR-STAGE FOURTH-ORDER IMPLICIT RUNGE-KUTTA METHOD BASED ON LEGENDRE POLYNOMIALS APPROXIMATION

利用勒让德多项式逼近理论和高斯-洛巴托求积公式，构造了一个4级4阶的隐式Runge-Kutta方法．理论分析发现，该算法具有良好的稳定性一是A（Q）稳定的且α接近于90^0，是刚性稳定的且D值接近于0，几乎是A稳定的和五稳定的，并能有效求解刚性常微分方程初值问题，数值算例显示了该算法的有效性．

By using the Legendre polynomials approximation theory and Gauss-Lobatto quadrature formula, a four-stage fourth-order implicit Runge-Kutta method is presented. It is showed that the new algorithm has good stability properties in theoretical analysis, A（a） -stable and a is close to ninety degrees, and stiff stable and D is close to zero. It is almost A-stable and almost L-stable. The new method can solve stiff ordinary differential equations effectively. The numerical examples illustrate its effectiveness.