Unitary Integrators and Applications to Continuous Orthonormalization Techniques

Abstract

In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques. In this case, the matrix system has a cubic nonlinearity. Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: automatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are in fact the Gauss–Legendre point Runge–Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples.