On the numerical integration of orthogonal flows with Runge-Kutta methods

This paper deals with the numerical integration of matrix differential equations of type Y'(t) = F(t, Y(r))Y(t) where F maps, for all r, orthogonal to skew-symmetric matrices. It has been shown (Dieci et al., SIAM J. Numer. Anal. 31 (1994) 261-281; Iserles and Zanna, Technical Report NA5, Univ. of Cambridge, 1995) that Gauss-Legendre Runge-Kutta (GLRK) methods preserve the orthogonality of the flow generated by Y' =F(t, Y)Y whenever F(t, Y) is a skew-symmetric matrix, but the implicit nature of the methods is a serious drawback in practical applications. Recently, Higham (Appl. Numer. Math. 22 (1996) 217-223) has shown that there exist linearly implicit methods based on the GLRK methods with orders less than or equal to 2 which preserve the orthogonality of the flow. The aim of this paper is to study the order and stability properties of a class of linearly implicit orthogonal methods of GLRK type obtained by extending Higham's approach. Also two particular linearly implicit schemes with orders 3 and 4 based on the two-stage GLRK method that minimize the local truncation error are proposed. In addition, the results of several numerical experiments are presented to test the behaviour of the new methods. (C) 2000 Elsevier Science B.V. All rights reserved.