Finite Difference Scheme for the Ericksen-leslie Equation

Ericksen Leslie equation describes the time evolution of a spin vector and velocity in liquid crystals. This equation has following property: (i) the length preserving of a spin vector, (ii) the energy conservation or the dissipation property, (iii) the incompressibility of a velocity vector. In physics papers, the fourth order Runge-Kutta’s method is used for numerical analysis of some types of the liquid crystal model(ex. [6] etc.). However, it abandons the properties (i), (ii). Some schemes which have already been proposed as the mathematical study inherit (ii) and (iii). By these schemes, the property (i) is obtained approximately. For example, these are based on the penalization method(ex. [3]). In this paper, we construct the new implicit scheme for Ericksen-Leslie equation which is based on the MAC method and inherits above three properties. Especially, this scheme inherits the property (i) directly.