常微分方程的半隐Runge-Kutta法  

Semi-implicit Runge-Kutta method for ordinary differential equations

作　　者：沈笠 马和平[1] 

机构地区：[1]上海大学理学院,上海200444

出　　处：《应用数学与计算数学学报》2017年第3期303-315,共13页Communication on Applied Mathematics and Computation

基　　金：国家自然科学基金资助项目(11171209);上海市教委重点学科建设资助项目(J50101)

摘　　要：针对某些非线性常微分方程,提出一种算子分裂半隐Runge-Kutta方法,对于非线性部分采用显式计算,对于刚性强的线性部分采用隐式处理.给出了格式的推导,分析了绝对稳定性,并证明了半隐二阶格式的收敛性.相比于显式Runge-Kutta法,半隐格式计算量相近,但改进了稳定性,数值结果显示了方法的合理性和有效性.最后,将算子分裂半隐Runge-Kutta方法应用于数值求解Zakharov偏微分方程组.In this paper, a splitting semi-implicit Runge-Kutta scheme is proposed for some nonlinear ordinary differential equations. The nonlinear part is computed explicitly, but the linear part of strong stiffness is treated implicitly. The scheme is derived, the absolute stability is analysed, and the convergence of the semi-implicit second-order Runge-Kutta is proved. Compared with the explicit Runge-Kutta method, the semi-implicit scheme costs nearly the same computing time but is of better stability. Numerical results show that the method is reasonable and effective. Finally, the semi-implicit Runge-Kutta scheme is applied to numerical solutions of the Zakharov equations.

关 键 词：RUNGE-KUTTA法 算子分裂 半隐格式 

分 类 号：O241.81[理学—计算数学]