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(t))Y(t) where F maps, for all t, 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 ⩽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 Highamʹs 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.