New approximate method for the Allen–Cahn equation with double-obstacle constraint and stability criteria for numerical simulations

Abstract

In a numerical study, we consider the Allen–Cahn equation with a double-obstacle constraint. The constraint is a multivalued function that is provided by the subdi erential of the indicator function on a closed interval. Therefore, performing a numerical simulation of our problem poses diffculties. We propose a new approximate method for the constraint and demonstrate its validity. Moreover, we present stability criteria for the standard forward Euler method guaranteeing stable numerical experiments when using the approximating equation.

Keywords

• Allen-Cahn equation| constraint| double obstacle| stability| subdi erential| numerical simulations