| 小波Galerkin法在非线性分岔问题求解中的应用 |
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| 引用本文： | 张磊,唐从刚,王德全,刘冰.小波Galerkin法在非线性分岔问题求解中的应用[J].应用数学和力学,2021,42(1):27-35. |
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| 作者姓名： | 张磊 唐从刚 王德全 刘冰 |
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| 作者单位： | 1国防科技大学 空天科学学院, 长沙410073;2湖北三江航天集团 红阳机电有限公司, 湖北 孝感 432000 |
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| 基金项目： | 湖南省自然科学基金（2019JJ50735） |
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| 摘 要： | 通过一个典型的Bratu问题，研究了小波Galerkin法（WGM）在非线性分岔问题求解方面的应用.首先，利用基于Coiflet的小波Galerkin法，对一维和二维Bratu方程进行离散；然后针对单参数问题，推导了追踪解曲线的伪弧长格式和直接计算极值型分岔点的扩展方程；针对双参数问题，推导了追踪稳定边界的伪弧长格式和求解尖点型分岔点的扩展方程.数值结果表明，基于小波Galerkin法的非线性分岔计算不仅具有更高的计算精度，而且能够有效地捕捉双参数分岔问题的折迭线和尖点突变曲面.该算例展示了基于小波Galerkin法的数值分岔计算的具体过程及其求解多参数分岔问题复杂行为的应用潜力.
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| 关 键 词： | 小波Galerkin法 分岔计算 双参数问题 尖点突变 |
| 收稿时间： | 2020-03-25 |
Application of the Wavelet Galerkin Method to Solution of Nonlinear Bifurcation Problems |
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| Affiliation: | 1College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, P.R.China;2Hubei Sanjiang Aerospace Hongyang Electromechanical Co. Ltd., Xiaogan, Hubei 432000, P.R.China |
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| Abstract: | Application of the wavelet Galerkin method to solution of nonlinear bifurcation problems was studied through a typical Bratu problem. Firstly, 1D and 2D Bratu equations were discretized with the Coiflet based wavelet Galerkin method, then both the pseudo arc length scheme for tracing solution curves and the extended equations for calculating limit bifurcation points were derived in the case of 1 parameter Bratu problems, similarly both the pseudo arc length scheme for tracing solution surfaces and the extended equations for solving cusp bifurcation points were also derived in the case of 2 parameter Bratu problems. Numerical results show that, the wavelet Galerkin method not only has higher accuracy during bifurcation point calculation, but also is capable of capturing fold lines and cusp catastrophe quantitatively in the case of 2 parameter bifurcation problems. This example exhibits the specific procedure of numerical bifurcation analysis based on the wavelet Galerkin method and demonstrates its potential for capturing complex bifurcation behaviors of multi parameter problems. |
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