alternating direction explicit method for a nonlinear model in finance

In this article, at first standard linear black-scholes model and then some nonlinear black-scholes models will be considered and thereupon alternating direction explicit (ade) method is applied firstly for solving the standard black-scholes model and then for barles and soner model which is one of the most complete and comprehensive nonlinear black-scholes models. furthermore, the stability of this method has been considered and its accuracy will be compared with other numerical methods such as finite difference methods. since in solving nonlinear black-scholes models by the ade methods, we need to solve only some scalar nonlinear equations instead of a full nonlinear system of equations that we should solve in implicit methods, so this method can be a suitable choice for solving such models.