Numerical Continuation for Nonlinear Problems with Symmetry Structures

对称结构非线性问题的数值延拓。Allgower本调查涉及的研究和发展numericalcontinuation方法，本质上利用对称性。将最近发展起来的利用线性方程组解的对称性的方法与数值延拓方法相结合，将为在跟踪分支的同时检测对称模式破缺分支，以及跟踪二次和多次分支的分支提供一种实用而有效的工具。将要研究和发展的延拓方法将集中于处理大型方程组，如偏微分方程组和积分方程组的离散化. 几种数值处理分叉的技术将被研究。其中包括中心流形理论的正规形式方程和Lyapunov-Schmidt约化。利用对称约化方法，得到了块对角化，极大地方便了分岔分析。在被处理的应用程序将是：非线性椭圆方程，如那些whichmodel屈曲的外壳对称，方程spheralconvection模型行星大气，Ginzburg-Landau方程建模的动态行波，和非线性边界积分方程是通过边界elementmethod离散。后一类积分方程自然地产生于外边界值问题.预计将开发的程序将通过匿名ftp以与研究者的数值延拓和对称约化方法程序相同的方式传播。大型结构系统的数值延拓，将在本研究中进行研究，具有广泛的重要应用领域。特别是，数值延拓可以用来计算模拟的变化，固有的控制参数引起的物理系统的定性变化。这些参数可能代表载荷、速度、温度、粘度、纵横比等。确定和分析控制参数的临界值对科学家和工程师具有重要意义，因为在这些值下会发生决定性的现象：稳定性可能会丧失，结构可能会断裂或共振，发生点火，大气流动模式发生变化等。从数学上讲，物理模型中控制参数的临界值用分歧点来表征。数值延拓方法可以用来检测和计算分歧点。研究将集中在具有几何对称结构的星系中发生的现象。对这种结构的考虑增强了科学理解，但也可以用于更精确和有效的计算。即将开发的计算机可视化程序将帮助研究人员和学生模拟、研究和洞察物理现象在参数变化下的模式变化。

DMS 9870274AbstractNumerical Continuation for Nonlinear Problems with Symmetry StructuresKurt Georg and Eugene L. Allgower This investigation deals with the study and development of numericalcontinuation methods which intrinsically exploit symmetries. Theintegration of techniques for exploiting symmetry in the solution oflinear systems, which have recently been developed by theinvestigators, into a numerical continuation method will provide apractical and efficient tool for detecting symmetry pattern breakingbifurcations while tracing a branch, and for following bifurcatingbranches for secondary and multiple bifurcations. The continuationmethods which will be studied and developed will concentrate upon thehandling of large systems of equations such as those stemming fromdiscretizations of systems of partial differential equations andintegral equations. Several techniques for numerically handlingbifurcations will be investigated. These include the normal formequations of center manifold theory and the Lyapunov-Schmidt reduction. By applying symmetry reduction methods developed by theinvestigators, block diagonalizations are obtained which greatlyfacilitate the bifurcation analysis. Among the applications to betreated will be: nonlinear elliptic equations such as those whichmodel the buckling of shells with symmetries, equations for sphericalconvection which model planetary atmospheres, Ginzburg-Landauequations modelling the dynamics of travelling waves, and nonlinearboundary integral equations which are discretized via boundary elementmethods. Integral equations of the latter kind arise in a natural wayfrom exterior boundary value problems. It is anticipated that theprograms which will be developed will be disseminated via anonymousftp in the same manner as the investigator's programs for numericalcontinuation and symmetry reduction methods have been done.Numerical continuation for large structured systems, which will bestudied in this investigation, has a broad field of importantapplications. In particular, numerical continuation can be used tocomputationally simulate how the variation of inherent controlparameters causes qualitative changes in a physical system. Suchparameters may represent loading, speed, temperature, viscosity,aspect ratio, etc. The determination and analysis of critical valuesof control parameters is of great significance to the scientist andengineer, because at these values decisive phenomena occur: stabilitycan be lost, structures can break or go into resonance, ingnitiontakes place, atmospheric flow patterns change, etc. Mathematically,critical values of control parameters in physical models arecharacterized by bifurcation points. Numerical continuation methodscan be used to detect and calculate bifurcation points. Theinvestigation will concentrate on phenomena taking place in regimeswhich have geometric symmetry structures. The consideration of suchstructures enhances the scientific understanding, but can also beexploited for more precise and efficient calculations. The computervisualization programs which will be developed will assist researchersand students to simulate, study, and gain insights into the patternchanges of physical phenomena under the variation of parameters.