求解半线性特征值问题的搜索延拓法

A Search Extension Method for Solving Semilinear Eigenvalue Problems

设计求解半线性椭圆特征值问题的改进型搜索延拓法(SEM),旨在实现以稳定方式计算多特征对的目标.该方法首先利用与模型问题对应的线性特征值问题的特征基的线性组合来搜索多特征对的初值;接着,适当增加特征基个数以获得更好的初值;然后,结合插值系数技巧与Legendre-Galerkin谱方法来离散模型问题,导出一个形式简单的非线性代数方程组,使得在每步牛顿迭代中更新雅可比矩阵只需计算一个对角矩阵;最后,用数值延拓法求解每个初值对应的特征对.该算法计算量小、谱精度高且易于实现.一类立方非线性特征值问题多特征对的丰富数值结果表明了方法的有效性,并展现出一些有趣的性质,包括特征对的分布规律,这些性质还有待证明.

A modified search extension method(SEM)for solving semilinear elliptic eigenvalue problems is designed to realize the goal of computing multiple eigenpairs by a stable way.This method first searches for initial guesses of multiple eigenpairs by the linear combination of eigenbases of the linear eigenvalue problem corresponding to the model problem.Next,appropriately increase the number of eigenbases to obtain better initial guesses.Then,combining the interpolation coefficient technique with the Legendre-Galerkin spectral method to discrete the model problem,a simple form of nonlinear algebraic equations is derived.So that only one diagonal matrix needs to be calculated for updating the Jacobian matrix in each Newton iteration.Finally,employ the numerical continuation method to find the eigenpair corresponding to each initial guess.The proposed algorithm is low in computational cost,spectrally accurate,and easy to implement.Rich numerical results of multiple eigenpairs for a class of cubic nonlinear eigenvalue problems indicate the effectiveness of the method and show some interesting properties including the distribution laws of eigenpairs,which are open to be proved.