LAUSR

A new finite difference pair is produced in this paper, for the first time in the literature. The characteristics of the new finite diffence pair are: (1) is of symmetric two-step, (2) is four-stages, (3) is of tenth-algebraic order, (4) the production of the pair is based on the following approximations for the layers: first and second layer are approximated on the point $$x_{n-1}$$xn-1, third layer is approximated on the point $$x_{n}$$xn and finally fourth layer is approximated on the point $$x_{n+1}$$xn+1, (5) has vanished the phase-lag and its first and second derivatives, (6) has excellent stability properties for all type of problems, (7) has an interval of periodicity equal to $$\left( 0, \infty \right) $$0,∞. We present for the new obtained finite difference pair a full theoretical analysis. The effectiveness of the new developed finite difference pair is proved by its application on systems of coupled differential equations arising from the Schrödinger equation.