【作者】 岳超；

【导师】 肖爱国；

【作者基本信息】 湘潭大学 ， 计算数学， 2008， 硕士

【摘要】 刚性问题是一类特殊的微分方程初值问题，具有广泛的应用背景．而Runge-Kutta方法是一种求解微分方程初值问题的常用的方法．对于刚性问题，我们一般采用隐式Runge-Kutta方法计算，这样我们可以得到高精度的数值解，但是要以很大的计算量为代价．因此，我们经常采用对角隐式Runge-Kutta方法来计算，以减少计算量．近年来，越来越多的学者对可分为两部分甚至更多的部分刚性问题产生了浓厚的兴趣．对于这类刚性问题的求解，为了减少计算量，我们采用叠加Runge-Kutta方法：对于刚性部分我们用隐式的Runge-Kutta方法；对于非刚性部分可用显式的Runge-Kutta方法．本文就是用叠加Runge-Kutta方法求解上述刚性问题，证明了代数稳定且对角稳定的叠加Runge-Kutta方法关于K_0,0类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件，也证明了弱代数稳定且对角稳定（或ANS-稳定）的叠加Runge-Kutta方法关于K_0,ω类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件．同时也证明了分数步方法Runge-Kutta方法关于K_0,0类、K_0,ω类初值问题的B-收敛性．最后证明了（θ，（?），（?））-代数稳定且对角稳定（或ANS-稳定）的叠加Runge-Kutta方法关于K_σ,τ类初值问题的最佳B-收敛阶不低于级阶，并获得了方法的最佳B-收敛阶比级阶高一的充分条件．同时也证明（θ，（?）,（?））-代数稳定的叠加Runge-Kutta方法关于K_φ,（?）类初值问题的单调性和分数步方法Runge-Kutta方法关于K_σ,τ类初值问题的B-收敛性．更多还原

【Abstract】 The stiff problems is a class of special initial-value problems for differential equations and have the widespread application background. But Runge-Kutta methods are used usually to solve the initial-value problems for differential equations. We often solve the stiff problems with the implicit Runge-Kutta methods. Although we can obtain the high-accuracy numerical solution, we pay large computational cost. In order to reduce the computational cost, we often use the diagonally-implicit Runge-Kutta methods to solve the initial-value problems. In recent years, much interest has been devoted to numerical integration of nonlinear stiff problems defined by operator that may be decomposed into a sum of two or more parts. To solve this class of stiff problems, we use additive Runge-Kutta methods. It is common to combine an implicit Runge-Kutta methods for stiff parts with explicit Runge-Kutta methods for nonstiff parts.In this article, we use the additive Runge-Kutta methods to solve the above stiff problems. We show that the order of optimal B-convergence of algebraically-stable and diagonally-stable additive Runge-Kutta methods is equal at least to the stage order for K_0,0 class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We also show that the order of optimal B-convergence of algebraically-stable and diagonally-stable（ANS-stable） additive Runge-Kutta methods is equal at least to the stage order for K_0,ω class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We show the optimal B-convergence of fractional step Runge-Kutta methods for K_0,0、K_0,ω class initial value problems. At last, we show that the order of optimal B-convergence of （θ, （p|-）, （q|-））-algebraically-stable and diagonally-stable（ANS-stable） additive Runge-Kutta methods is equal at least to the stage order for K_σ,τ class initial value problems, and provide some sufficient conditions under which the order of optimal B-convergence is one higher than stage order. We show that the monotonicity of （θ, （p|-）, （q|-））-algebraically-stable and the optimal B-convergence of fractional step Runge-Kutta methods for K_Φ,φ class initial value problems.更多还原