Modified Verlet Method Involving Second-Order Mid-Point Rule Applied to Balls Falling in One-Dimensional Potentials
Hidenori Yasuda 1*1 Department of Mathematics, Faculty of Science, Josai University, 1-1 Keyakidai, Sakado, Saitama, Japan.
Received 16 September 2009; Accepted (in revised version) 31 March 2010
Available online 26 October 2010
A modified Verlet method which involves a kind of mid-point rule is constructed and applied to the one-dimensional motion of elastic balls of finite size, falling under constant gravity in space and then under the chemical potential in the interface region of phase separation within a two-liquid film. When applied to the simulation of two balls falling under constant gravity in space, the new method is found to be computationally superior to the usual Verlet method and to Runge--Kutta methods, as it allows a larger time step for comparable accuracy. The main purpose of this paper is to develop an efficient numerical method to simulate balls in the interface region of phase separation within the two-liquid film, where the ball motion is coupled with two-phase flow. The two-phase flow in the film is described via shallow water equations, using an invariant finite difference scheme that accurately resolves the interface region. A larger time step in computing the ball motion, more comparable with the time step in computing the two-phase flow, is a significant advantage. The computational efficiency of the new method in the coupled problem is demonstrated for the case of four elastic balls in the two-liquid film.
AMS subject classifications: 65L12, 65M06, 65P10
Key words: Verlet method, falling balls, first return map, phase separation, shallow water equations.
Email: email@example.com (H. Yasuda)