L. Dieci, R.D. Russell and E.S. Van Vleck,

``Unitary Integrators and Applications to Continuous Orthonormalization Techniques,''

(1994) SIAM J. Numer. Anal. 31 pp. 261-281.

ABSTRACT

In this paper we address the issue of integrating matrix differential systems
whose solutions are unitary matrices. Such systems have skew-hermitian
coefficient matrices in the linear case and have a related structure in the
nonlinear case. These skew systems arise in a number of applications, and our
own interest originates from application to continuous orthogonal decoupling
techniques. In this case, the matrix system has a cubic nonlinearity. Here, we
are interested in studying numerical integration schemes which compute a
unitary approximate solution for all step-sizes. We see that these schemes can
be characterized as being of two classes: {\it automatic} and {\it 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 RK schemes. The second class of schemes is created by projecting
approximations computed by an arbitrary scheme into the set of unitary
matrices. In our 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.