Numerical procedures based on Runge-Kutta methods for solving isospectral flows Original Research Article

This paper deals with the numerical solution of the Lax system L′ = [B(L),L], L(0) = L0 (∗), where L0 is a constant symmetric matrix, B(·) maps symmetric matrices into skew-symmetric matrices, and [B(L),L] is the commutator of B(L) and L. Here two different procedures, based on the approach recently proposed by Calvo, Iserles and Zanna (the MGLRK methods), are suggested. Such an approach is a computational form for the Flaschka formulation of (∗). Our numerical procedures consist in solving (∗) by a Runge-Kutta method, then, a single step of a Gauss-Legendre Runge-Kutta (GLRK) method may be applied to the Flaschka formulation of (∗). In the first procedure we compute the approximation of the Lax system by a continuous explicit RK method, instead, the second procedure computes the approximation of the Lax system by a GLRK method (the same method used for the Flaschka system). The computational costs have been derived and compared with the ones of the MGLRK methods. Finally, several numerical tests and computational comparisons will be shown.