求解微分方程初值问题的一种弧长法

基金项目: 国家自然科学基金

Arc-Length Method for Differential Equations

1.
Department of Mechanics and Engineering Science, Peking University, Beijing 100871, P. R. China;
2.
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, P. R. China

摘要: 对于连续介质力学问题中导出的微分方程初值问题,常常具有解奇异性,如不连续、Stif性质或激波间断.本文通过在相应空间,引入一个或数个弧长参数变量,克服解的奇异性.对于常微分方程组引入弧长参数变量后,奇异性得以消除和削弱,应用一般的解常微分方程组的方法(如Runge-Kuta法)求解.对于偏微分方程引入弧长参数变量后,在相应的空间离散成常微分方程组,用解奇异性常微分方程组相同的方法即可求解.本文给出了两个算例.
- 微分方程 /
- 数值方法 /
- 弧长法 /
- Stiff方程 /
- Burgers’方程
Abstract: A kind arc-length method is presented to solve the ordinary differential equations(ODEs) with certain types of singularit y as stiff property or discontinuity on continuum problem. By introducing one or two arc-length parameters as variables, the differential equations with singularity are transformed into non-singularity equations, which can be solved by usual methods. The method is also applicable for partial differential equations(PDEs), because they may be changed into systems of ODEs by discretization. Two examples are given to show the accuracy, efficiency and application.
- differential equation /
- numerical method /
- arc-length method /
- stiff equation /
- Burgers’ equation

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