LAUSR

In this paper and for the first time in the literature, a new four-stages symmetric two-step finite difference pair with optimized phase and stability properties is introduced. The new scheme has the following characteristics:1.is of symmetric type,2.is of two-step algorithm,3.is of four-stages,4.is of tenth-algebraic order,5.the approximations which produces the new finite difference pair are the following:An approximation developed on the first layer on the point $$x_{n-1}$$xn-1,An approximation developed on the second layer on the point $$x_{n-1}$$xn-1,An approximation developed on the third layer on the point $$x_{n}$$xn and finally,An approximation developed on the fourth (final) layer on the point $$x_{n+1}$$xn+1,6.it has eliminated the phase-lag and its first, second, third and fourth derivatives,7.it has optimized stability properties,8.is a P-stable methods since it has an interval of periodicity equal to $$\left( 0, \infty \right) $$0,∞. For the new finite difference scheme we describe an error and stability analysis. The examination of the efficiency of the new obtained finite difference pair is based on its application on systems of coupled differential equations arising from the Schrödinger equation.