The Isospectral Flow Methods And Their Applications

Isospectral flows are extensively applied to many fields such as molecular dy-namics, micro-magnetics, and linear algebra, and they has aroused many interests in recent years. Such flows are created by a matrix differential equation L=[B(L),L],L(0)=L0, where Lo is a given symmetric matrix,B(L)is skew-symmetric for all L, and [B(L),L]=B(L)L-LB(L)is the commutator of B and L.The choices of B characterizes the different dynamics of isospectral flows.In this thesis, we develop a new isospectral flow methods for the Toda flow and compare several isospectral flow methods on their ability to preserving the energy conservation law. The main work of this thesis is as follows:1. We briefly review the basic theory of isospectral flows and several classical examples of isospectral flows. Connection of the isospectral with the QR algorithm are also presented.2. We study the classical isospectral flow methods, such as the modified Gauss-Legendre Runge-Kutta (MGLRK) schemes and semi-explicit Isospectral Taylor meth-ods. Their numerical performance on energy preserving are compared.3. Based on the basic concepts of Lie groups method and the basic principles of RKMK method, a new and more effective isospectral flow method, namely the RK method with Cayley transform are proposed. Furthermore, we applied the new method to the Toda flow. Numerical results show that the new method is more efficient than the classical one and it also has good ability to preserve the energy.