毕业设计(论文)毕业设计题目:刚性系统的隐式RK方法学院:数理学院专业名称:信息与计算科学学号:201241210127学生姓名:丁楠指导教师:汪玉霞2016年05月15日毕业设计(论文)摘要本文主要介绍单步隐式Runge–Kutta方法,简要的介绍了Gauss型隐式Runge–Kutta方法、Radau型隐式Runge–Kutta方法和Lobatto型隐式Runge–Kutta方法。并利用这些基本的隐式Runge–Kutta方法来对刚性方程组进行数值求解,并将隐式Runge–Kutta方法与显式经典Runge–Kutta方法求解的结果进行对比,说明两种数值解法的优缺点。关键词:刚性系统隐式Runge–Kutta方法单步方法Newton迭代法AbstractThispapermainlyintroducestheImplicitRunge-KuttaMethodsandasimpledescriptionofGaussimplicitRunge-Kuttamethod,RadauimplicitRunge-KuttamethodandLobattoimplicitRunge-Kuttamethod.ThesebasicImplicitRunge-Kuttamethodsareusedtosolvethestiffequations.TheseimplicitRunge-KuttamethodsiarecomparedwiththeclassicalexplicitRunge-Kuttamethod.Thispaperexplaintheadvantagesanddisadvantagesofthetwokindofnumericalmethods.Keywords:StiffsystemImplicitRunge-KuttamethodOnestepmethodNewtoniterativemethod毕业设计(论文)目录1.绪论...............................................................................................................................11.1刚性方程.............................................................................................................11.2隐式RK方法的研究意义...................................................................................21.3RK方法的研究现状...........................................................................................32.单步RK方法的收敛性和稳定性................................................................................52.1单步RK方法的收敛性.......................................................................................52.2单步RK方法的稳定性.......................................................................................63.三类隐式RK方法........................................................................................................83.1引言.....................................................................................................................83.2Gauss型隐式RK方法........................................................................................93.3Radau型隐式RK方法.....................................................................................103.4Lobatto型隐式RK方法...................................................................................114隐式RK方法的实现..................................................................................................134.1非线性系统的改进............................................................................................134.2简化的Newton迭代法.....................................................................................135数值实验与结果分析.................................................................................................15参考文献........................................................................................................................18附录................................................................................................................................21毕业设计(论文)毕业设计(论文)1.绪论1.1刚性方程对于一般的线性常系数系统y'=Ay+φ(t)A为m×m的矩阵,特征值为λi(i=1,2,⋯,m)。定义1[23]若一个系统满足(1)ℜλi<0,i=1,2,⋯,m(2)maxi|ℜλi|/mini|ℜλi|=R≫1其中...