一类复合法求解 Burgers 型方程 Alternative Title A composition method for solving Burgers-type Equations 苗会会 Thesis Advisor 周宇斌 2014-05-29 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 龙格-库塔法 辛自伴随的龙格-库塔法 复合法 KdV-Burgers方程 Burgers 方程组 Abstract 龙格-库塔法是一类经典的数值方法, 被广泛的应用在求解各种微分方程中.本文将龙格-库塔法按照一定的规则进行复合, 得到了两种精度更高的复合型的数值方法, 并通过一些数值实验验证了该复合法的有效性和可行性. 首先, 基于显式的龙格-库塔法构造了一类复合法, 并将该方法应用到了KdV-Burgers 方程的求解中, 该方程是利用Chebyshev 谱配置法将空间变量进行离散得到的一个关于时间变量t 的常微分方程. 数值实验显示该方法与Sinc函数法相比具有更高的精度. 此外, 还将该方法分别应用到(1+1)和(2+1)维耦合Burgers 方程组进行求解中, 通过具体的数值实验对比发现本文的复合法要比没有经过复合的龙格-库塔法的精度高至少一个数量级, 说明本文的复合法具有一定的优势. 然后, 基于辛自伴随的龙格-库塔法构造一种复合法, 并将该方法应用到(1+1)维Burgers 方程的求解中, 通过数据对比发现本文构造的复合法比未复合的方法具有很好的计算稳定性, 在某些特定的条件下精确度更高. 最后, 对(1+1)维耦合的Burgers 方程组的空间变量进行分区间离散, 数值实验说明该方法能够减小绝对误差. Other Abstract The Runge-Kutta method, as a kind of classical numerical methods, is widely used in solving a variety of differential equations. In this work, we composite the Runge-Kutta method according to certain rules and get two new higher precision composition methods. Furthermore, numerical experiments are presented to demonstrate the efficiency and feasibility of the composition method. First, we construct a composite method based on the explicit Runge-Kutta method. The new composition method is used to solve KdV-Burgers equation, an ODE about time variable t whose space variable is discretized by Chebyshev collocation method. Numerical experiments depict that the composition method has higher accuracy than the Sinc method. Besides, this method is used to solve (1+1) and (2+1) dimensional coupled Burgers equations. Numerical experiments show that the composited method has higher precision of at least one orders’ magnitude than the Runge-Kutta method that are not composited. Then, a composite method based on the self-adjoint Runge-Kutta method is constructed. We use this method to solve the (1+1)-dimensional Burgers equation. Numerical experiments show that this composited method has high computational stability and more accurate than the Runge-Kutta method that are not composited under certain conditions. The last, the (1+1) dimensional coupled Burgers equations’s space variable is discretized by partitioned. Numerical experiment depict that the method can reduce the absolute error. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224494 Collection 数学与统计学院 Recommended CitationGB/T 7714 苗会会. 一类复合法求解 Burgers 型方程[D]. 兰州. 兰州大学,2014.
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