On symplectic and isospectral integration of the stationary Landau-Lifshitz (Neumann oscillator) equation

Abstract

In this paper we discuss numerical integration of the stationary Landau–Lifshitz
(LL) equation. Using a Lax pair representation of the LL equation, we propose an isospectral algorithm that preserves the geometric structure of the system. The algorithm computes a discrete flow of a pair of matrices satisfying Lax–type equations and projects the flow on the phase space of the system. Since the stationary LL equation is equivalent to an integrable Hamiltonian system on the cotangent bundle of the unit sphere, we show that it can be also integrated by a symplectic method for constrained Hamiltonian systems. Comparison of the two methods demonstrates that they are similar in terms of accuracy
and stability over long–time integration, but the isospectral method is much faster since it avoids solving a system of nonlinear equations required at each iteration of the symplectic algorithm.