Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods
The paper derives and investigates multidimensional exponential fitting modified Runge-Kutta-Nyström (MEFMRKN) methods for a system of oscillatory second order differential equations of the form \[ \begin{cases} q'(t)+Mq(t)=f(q(t)),\;t\in[0,T],\\ q(0)=q_0,\;q'(0)={q_0}' \end{cases} \tag{1} \] where \(M\) is a \(d\times d\) symmetric and positive semi-definite matrix implicitly containing the frequencies of the problem, \(q:{\mathbb R}\rightarrow{\mathbb R}^d\) is the solution of the system and \(f(q)\) is the gradient of a real-valued function \(U(q)\) whose second derivatives are continuous. \noindent Taking into account the fact that the oscillatory system is a separable Hamiltonian system with Hamiltonian \(H(p,q)=\frac{1}{2}p^Tp+\frac{1}{2}q^TMQ+U(q),\) where \[ \begin{cases} q'(t)=p\\ p'=-\nabla U(q)-Mq\\ q(0)=q_0,\;p(0)=p_0 \end{cases} \] one derives the symplecticity conditions for the MEFMRKN methods. The first section is an introduction in the nature. Through the second section one achieves the multidimensional exponential fitting modified RKN or MEFMRKN schemes for the multidimensional oscillatory system (1), showing the relationship between MEFMRKN and RKN integrators. The third section presents and proves the symplecticity conditions for multidimensional modified RKN methods and then for MEFMRKN methods. Taking into account the fact that the explicit symplectic integrators do not require the solution of large and complicate systems of nonlinear algebraic or transcedental equations when solving multidimensional problems, the fourth section presents two explicit symplectic MEFMRKN methods: a two-stage explicit symplectic MEFMRKN method and a three-stage explicit symplectic MEFMRKN method. The efficiency of the presented methods is demonstrated in the fifth section, by five numerical experiments corresponding to two kinds of the problems. The main conclusions are within the last section.
scientific article; zbMATH DE number 6109252
scientific article