Novel improved multidimensional Störmer–Verlet formulas with applications to four aspects in scientific computation

This paper presents two novel improved multidimensional Störmer–Verlet formulas with four applications to time-independent Schrödinger equations, wave equations, orbital problems and the problem of Fermi, Pasta & Ulam. For solving the system of second-order ordinary differential equations y″+My=f(t,y)y″+My=f(t,y) with M∈Rm×mM∈Rm×m, the multidimensional ARKN methods (adapted Runge–Kutta–Nyström methods) were formulated by Wu et al. (2009) [1]. Very recently, the multidimensional ERKN methods (extended Runge–Kutta–Nyström methods) were proposed by Wu et al. (2010) [26]. Both the ARKN methods and the ERKN methods perform numerically much better than the classical Runge–Kutta–Nyström methods due to the use of the special structure of the equation brought by the linear term MyMy. Based on the two kinds of multidimensional schemes, we derive two novel improved multidimensional Störmer–Verlet formulas, which are shown to be symplectic and of order two. Each new formula is a blend of existing trigonometric integrators and symplectic integrators. Meantime, the symplecticity conditions for the one-stage explicit multidimensional ARKN methods are presented. Stability and phase properties of the two improved formulas are analyzed. Numerical experiments demonstrate that the two improved multidimensional Störmer–Verlet formulas are more efficient than the classical Störmer–Verlet formula and the two other improved Störmer–Verlet methods appeared in the literature. In particular, when applied to a Hamiltonian system, the two symplectic improved multidimensional Störmer–Verlet formulas preserve well the Hamiltonian in the sense of numerical approximation, and have better accuracy than the classical Störmer–Verlet formula and the two other improved Störmer–Verlet methods with the same computational cost.