One step semi-explicit methods based on the Cayley transform for solving isospectral flows

This note deals with the numerical solution of the matrix differential system Y′ = B(t,Y), Y], Y(0) = Y₀, t ⩾ 0, where Y₀ is a real constant symmetric matrix, B maps symmetric into skew-symmetric matrices, and B(t,Y),Y] is the Lie bracket commutator of B(t,Y) and Y, i.e. B(t,Y),Y] = B(t,Y)Y − YB(t,Y). The unique solution of (1) is isospectral, that is the matrix Y(t) preserves the eigenvalues of Y₀ and is symmetric for all t (see 1, 5]). Isospectral methods exploit the Flaschka formulation of (1) in which Y(t) is written as Y(t) = U(t)Y₀U^T(t), for t ⩾ 0, where U(t) is the orthogonal solution of the differential system U′ = B(t, UY₀U^T)U, U(0) = I, t ⩾ 0, (see 5]). Here a numerical procedure based on the Cayley transform is proposed and compared with known isospectral methods.