In this paper, for the first time in the literature, we introduce a new five–stages symmetric two–step method with improved phase and stability properties. The characteristics of the new non linear two–step method are: (1) it is of symmetric type, (2) it is of two–step finite difference pair, (3) it is of five–stages, (4) it is of twelfth–algebraic order, (5) it has vanished the phase–lag and its first and second derivatives, (6) it has improved stability properties for the general problems, (7) it is a P-stable method since it has an interval of periodicity equal to $$\left( 0, \infty \right) $$ 0,∞ , (8) for the development of the new scheme, the following approximations are used: (a) an approximation developed on the first layer on the point $$x_{n-1}$$ xn-1 , (b) an approximation developed on the second layer on the point $$x_{n-1}$$ xn-1 , (c) an approximation developed on the third layer on the point $$x_{n-1}$$ xn-1 , (d) an approximation developed on the fourth layer on the point $$x_{n}$$ xn and finally, (e) an approximation developed on the fifth (final) layer on the point $$x_{n+1}$$ xn+1 . For the new developed method we give a full theoretical analysis (error and stability analysis). The efficiency of the new five–stages symmetric two–step method is examined by applying it on the numerical solution of systems of coupled differential equations arising from the Schrödinger equation.