Variable step-size techniques in continuous Runge-Kutta methods for isospectral dynamical systems

In this paper we consider numerical methods for the dynamical system L′ = [B(L), L], L(0) = L0, (*) where L0 is a n × n symmetric matrix, [B(L), L] is the commutator of B(L) and L, and B(L) is a skew-symmetric matrix for each symmetric matrix L. The differential system is isospectral, i.e., L(t) preserves the eigenvalues of L0, for t ≥ 0. The matrix B(L) characterizes the flow, and for special B(·), the solution matrix L(t) tends, as t increases, to a diagonal matrix with the same eigenvalues of L0. In [11] a modification of the MGLRK methods, introduced in [2], has been proposed. These procedures are based on a numerical approximation of the Flaschka formulation of (*) by Runge-Kutta (RK) methods. Our numerical schemes (denoted by EdGLRKs) consist in solving the system (*) by a continuous explicit Runge-Kutta method (CERK) and then performing a single step of a Gauss-Legendre RK method, for the Flaschka formulation of (*), in order to convert the approximation of L(t) to an isospectral solution. The problems of choosing a constant time step or a variable time step strategy are both of great importance in the application of these methods. In this paper, we introduce a definition of stability for the isospectral numerical methods. This definition involves a potential function associated to the isospectral flow. For the class EdGLRKs we propose a variable step-size strategy, based on this potential function, and an optimal constant time step h in the stability interval. The variable time step strategy will be compared with a known variable step-size strategy for RK methods applied to these dynamical systems. Numerical tests will be given and a comparison with the QR algorithm will be shown.