Numerical integration methods for the double-bracket flow.
This paper is concerned with the numerical solution of the so called double-bracket matrix differential equation \( Y'(t)= [ [ Y(t), N ],] Y(t)], Y(0)= Y_0\) where \(N\) and \(Y_0 \) are \( n \times n \) constant real matrices with \(Y_0\) symmetric, and \( [A,B]= A B-B A\). Since the author is interested in geometric integrators and this equation is a typical example of isospectral flow, the proposed methods retain this property. Taking into account that \( Y(t)\) can be written as \( Y(t)= Q(t) Y_0 Q(t)^T \) with orthogonal \( Q(t)\) and also the representation \( Q(t)= \exp ( \Omega (t))\) with skew-symmetric \( \Omega (t)\) the author proposes a Magnus series expansion up to fourth order of \( \Omega (t)\). This order has been restricted to four to minimize the number of required commutators. A theoretical study of the convergence of the new method is presented that allows the author to include dependence on the matrix norm selected. Finally, some numerical experiments comparing the new method with other fourth order Lie-group Runge-Kutta (RK) integrator due to \textit{H. Munthe-Kaas} and \textit{B. Owren} [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 357, No. 1754, 957--981 (1999; Zbl 0956.65056)] which also preserves isospectrality and the classical RK4 method of order four are presented. As expected geometric integrators provide much better approximations to the eigenvalues than the classical RK4. Moreover the new method shows much better stability properties than the RK of Munthe-Kaas and Owren [loc. cit.].
geometric integration, Magnus expansions
double-bracket matrix differential equation
scientific article; zbMATH DE number 2062748