UNITARY INTEGRATORS AND APPLICATIONS TO CONTINUOUS ORTHONORMALIZATIONTECHNIQUES

In this paper the issue of integrating matrix differential systems who se solutions are unitary matrices is addressed. Such systems have skew -Hermitian coefficient matrices in the linear case and a related struc ture in the nonlinear case. These skew systems arise in a number of ap plications, and interest originates from application to continuous ort hogonal decoupling techniques. In this case, the matrix system has a c ubic nonlinearity. Numerical integration schemes that compute a unitar y approximate solution for all stepsizes are studied. These schemes ca n be characterized as being of two classes: automatic and projected un itary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are i n fact the Gauss-Legendre point Runge-Kutta (Gauss RK) schemes. The se cond class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analy sis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and impl ementation issues are considered, and the methods are tested on a numb er of examples.