Numerical Continuation for Nonlinear Problems with Symmetry Structures

DMS 9870274对称结构非线性问题的数值延拓这项研究涉及本质上利用对称性的数值延拓方法的研究和发展。将研究人员最近开发的利用线性系统解的对称性的技术集成到数值连续方法中，将为在跟踪分支的同时检测对称模式的分叉以及跟踪二次和多次分叉的分叉提供实用和有效的工具。将研究和发展的延拓方法将集中于处理大型方程组，如由偏微分方程组和积分方程组的离散化引起的方程组。我们将研究几种数值处理分叉的技术。其中包括中心流形理论的正规公式和Lyapunov-Schmidt约化。应用研究人员发展的对称性约化方法，得到了块对角化，这极大地方便了分叉分析。其中的应用包括：非线性椭圆型方程，如那些用来模拟具有对称性的壳的屈曲的方程，用来模拟行星大气的球面对流方程，用来模拟行波动力学的Ginzburg-Landau型方程，以及用边界元方法离散的非线性边界积分方程组。后一类积分方程组是由外边值问题自然产生的。预计将开发的程序将通过匿名ftp以与研究人员的数值延拓和对称性约化方法的程序相同的方式传播。大型结构系统的数值延拓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固有控制参数的变化如何引起物理系统的质的变化。这些参数可以表示载荷、速度、温度、粘度、展弦比等。控制参数临界值的确定和分析对科学家和工程技术人员具有重要意义，因为在这些临界值上会出现决定性的现象：可能失去稳定性，结构可能破裂或进入共振，振动发生，大气流型变化等。数值连续法可以用来检测和计算分叉点。研究将集中在具有几何对称结构的星系中发生的现象。对这种结构的考虑加强了对科学的理解，但也可以用于更精确和更有效的计算。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.