Comparison of second-order mixed symplectic integrator between semi-implicit Euler method and implicit midpoint rule

Abstract

When a Hamiltonian can be split into integrable and nonintegrable parts, the former part is solved analytically, and the latter one is integrated numerically by means of implicit symplectic integrators such as the first-order semi-implicit Euler method or the second-order implicit midpoint rule. These analytical and numerical solutions are used to construct a second-order mixed symplectic integrator with the semi-implicit Euler method and one with the implicit midpoint rule. A theoretical analysis shows that the Euler mixed integrator is inferior to the midpoint one in the sense of numerical stability. Numerical simulations of the circularly-restricted three-body problem also support this fact. It is further shown through numerical integrations of the post-Newtonian Hamiltonian of spinning compact binaries that the qualities of the Euler mixed integrator and the midpoint mixed method do depend on the type of orbits. Especially for chaotic orbits, the Euler mixed integrator often becomes unstable. In addition, the Euler mixed integrator has an advantage over the midpoint mixed method in computational efficiency, and is almost equivalent to the latter in the numerical accuracy if the two mixed integrators are stable. In spite of this, the midpoint mixed integrator is worth recommending for the study of the dynamics of post-Newtonian Hamiltonians of spinning compact binaries.

Application of the Störmer—Verlet-Like Symplectic Method to the Wave Equation

“…with 𝛼 = −1/12, 𝛽 = 1/24 and 𝛾 = 1/6. When the three algorithms act, respectively, on the discrete Hamiltonian (7), their difference schemes from a 𝑙th step to a (𝑙 + 1)th step are…”

Dynamics of cubic–quintic nonlinear Schrödinger equation with different parameters

“…[11][12][13] Up to now, it has been applied to many fields, such as the quantum mechanics, [14] molecular dynamics, [15] Bose-Einstein condensates, [16][17][18][19][20] solar system dynamics, and general relativity. [21][22][23][24][25] A fourth-order, threestage, symplectic integrator with small numerical error has been used, and a method of fast Lyapunov indicator has been applied in studying the dynamics of the system. [21,23] The Lyapunov exponents and the fast Lyapunov indicators are chaotic indicators independent of the dimensionality of phase space, and they are widely used in higher-dimensional systems to distinguish chaos from order.…”

“…[21][22][23][24][25] A fourth-order, threestage, symplectic integrator with small numerical error has been used, and a method of fast Lyapunov indicator has been applied in studying the dynamics of the system. [21,23] The Lyapunov exponents and the fast Lyapunov indicators are chaotic indicators independent of the dimensionality of phase space, and they are widely used in higher-dimensional systems to distinguish chaos from order. [24] For example, the fast Lyapunov indicator has been used in the three-body problem and compact binary systems to identify chaos.…”