2019, 40(6): 595-608.
doi: 10.21656/1000-0887.390249
Abstract:
Symplectic methods, including the generating function method, the symplectic Runge-Kutta (RK) method, the symplectic partitioned Runge-Kutta method, the multi-step method and so on, are applicable to Hamiltonian systems. They can preserve the symplectic structure in the phase space and the laws of the Hamiltonian system. But in the time domain, due to phase lags in the computing course, the RK methods and the symplectic methods have the same algebraic precision under the same algebraic order of schemes. After longtime computing, the numerical precision goes worse and worse in the time domain. To improve the precision, a new method combining the highly precise direct integration method with the symplectic difference scheme, called the HPD-symplectic method, was proposed. This method, proved to be symplectic, can preserve the symplectic structure. Moreover, the HPD-symplectic method can largely decrease the phase error in the time domain, and accordingly, improve the numerical precision even up to an error level of 10-13. For systems with mixed frequencies or rigid systems, the traditional symplectic methods can hardly work well, while the HPD-symplectic method can simulate the signals at both high and low frequencies well with large time steps but no additional computation cost. The results of numerical examples demonstrate the reliability and effectiveness of the proposed method.
