Symplectic-Structure-Preserving Uncertain Differential Equations

Uncertain differential equations are important mathematical models in uncertain environments. This paper investigates uncertain multi-dimensional and multiple-factor differential equations. First, the solvability of the equations is analyzed. The α-path distributions and expected values of solutions are given. Then, structure preserving uncertain differential equations, uncertain Hamiltonian systems driven by Liu processes, which possess a kind of uncertain symplectic structures, are presented. A symplectic scheme with six-order accuracy and a Yao-Chen algorithm are applied to design an algorithm to solve uncertain Hamiltonian systems. At last, numerical experiments are given to investigate four uncertain Hamiltonian systems, which highlight the efficiency of our algorithm.

Taking apart the dynamical clock

Context. In globular clusters (GCs), blue straggler stars (BSS) are heavier than the average star, so dynamical friction strongly affects them. The radial distribution of BSS, normalized to a reference population, appears bimodal in a fraction of Galactic GCs, with a density peak in the core, a prominent zone of avoidance at intermediate radii, and again higher density in the outskirts. The zone of avoidance appears to be located at larger radii the more relaxed the host cluster, acting as a sort of dynamical clock. Aims. We use a new method to compute the evolution of the BSS radial distribution under dynamical friction and diffusion. Methods. We evolve our BSS in the mean cluster potential under dynamical friction plus a random fluctuating force, solving the Langevin equation with the Mannella quasi symplectic scheme. This is a new simulation method that is much faster and simpler than direct N-body codes, but retains their main feature: diffusion powered by strong, if infrequent, kicks. Results. We compute the radial distribution of initially unsegregated BSS normalized to a reference population as a function of time. We trace the evolution of its minimum, corresponding to the zone of avoidance. We compare the evolution under kicks extracted from a Gaussian distribution to that obtained using a Holtsmark distribution. The latter is a fat-tailed distribution which correctly models the effects of close gravitational encounters. We find that the zone of avoidance moves outwards over time, as expected based on observations, only when using the Holtsmark distribution. Thus, the correct representation of near encounters is crucial to reproduce the dynamics of the system. Conclusions. We confirm and extend earlier results that showed how the dynamical clock indicator depends on dynamical friction and on effective diffusion powered by dynamical encounters. We demonstrated the high sensitivity of the clock to the details of the mechanism underlying diffusion, which may explain the difficulties in reproducing the motion of the zone of avoidance across different simulation methods.

Compact splitting symplectic scheme for the fourth-order dispersive Schrödinger equation with Cubic-Quintic nonlinear term

Combining symplectic algorithm, splitting technique and compact method, a compact splitting symplectic scheme is proposed to solve the fourth-order dispersive Schrödinger equation with cubic-quintic nonlinear term. The scheme has fourth-order accuracy in space and second-order accuracy in time. The discrete charge conservation law and stability of the scheme are analyzed. Numerical examples are given to confirm the theoretical results.

A flexible symplectic scheme for two-dimensional Schrödinger equation with highly accurate RBFS quasi-interpolation

Based on highly accurate multiquadric quasi-interpolation, this study suggests a meshless symplectic procedure for two-dimensional time-dependent Schr?dinger equation. The method is highorder accurate, flexible with respect to the geometry, computationally efficient and easy to implement. We also present a theoretical framework to show the conservativeness and convergence of the proposed method. As the numerical experiments show, it not only offers a high order accuracy but also has a good performance in the long time integration.