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谭述君,高强,钟万勰.Duhamel项的精细积分方法在非线性微分方程数值求解中的应用[J].计算力学学报,2010,27(5):752~758

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Duhamel项的精细积分方法在非线性微分方程数值求解中的应用

Applications of Duhamel term’s precise integration method in solving nonlinear differential equations

投稿时间：2008-09-27

DOI：10.7511/jslx20105002

中文关键词: 非线性 Duhamel积分精细积分方法 Taylor级数展开 Adams线性多步法刚性

英文关键词:nonlinear Duhamel integration precise integration method Taylor series expanding Adams linear multi-step method stiffness

基金项目:国家自然科学基金(10632030);高等学校博士学科点专项科研基金(20070141067)资助项目.

作者	单位
谭述君	大连理工大学工业装备结构分析国家重点实验室,大连 116023
高强	大连理工大学工业装备结构分析国家重点实验室,大连 116023
钟万勰	大连理工大学工业装备结构分析国家重点实验室,大连 116023

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中文摘要:

基于Duhamel项的精细积分方法,构造了几种求解非线性微分方程的数值算法。首先将非线性微分方程在形式上划分为线性部分和非线性部分,对非线性部分进行多项式近似,利用Duhamel积分矩阵,导出了非线性方程求解的一般格式。然后结合传统的数值积分技术,例如Adams线性多步法等,构造了基于精细积分方法的相应算法。本文算法利用了精细积分方法对线性部分求解高度精确的优点,大大提高了传统算法的数值精度和稳定性,尤其是对于刚性问题。本文构造的算法不需要对线性系统矩阵求逆,可以方便的考察不同的线性系统矩阵对算法性能的影响。数值算例验证了本文算法的有效性,并表明非线性系统的线性化矩阵作为线性部分是比较合理的选择。

英文摘要:

Several numerical algorithms for nonlinear differential equations are constructed based on Duhamel term’s precise integration method (PIM). Firstly the nonlinear differential equations are formally divided into linear and nonlinear parts and then the latter are approximated by polynomials. By applying the Duhamel integration, the general discrete forms for nonlinear differential equations are derived. Then the corresponding PIM-based algorithms are constructed by combining the traditional numerical integration techniques, such as Adams linear multi-step method, with PIM. Compared with the traditional algorithms, the new method integrates the linear part accurately by using PIM, and so improves the numerical precision and stability significantly, especially for stiff problems. Furthermore, the method proposed in this paper avoids the matrix inverse for the linear part and it is convenient to study effects of the linear matrix on the algorithm’s performance. Numerical experiments confirmed the validity of the proposed method and show that the linearization matrix of the nonlinear system is a good choice for the linear part.

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