New hybrid two-step method with optimized phase and stability characteristics

In this paper and for the first time in the literature, we develop a new Runge–Kutta type symmetric two-step finite difference pair with the following characteristics:the new algorithm is of symmetric type,the new algorithm is of two-step,the new algorithm is of five-stages,the new algorithm is of twelfth-algebraic order,the new algorithm is based on the following approximations:1.the first layer on the point $$x_{n-1}$$xn-1,2.the second layer on the point $$x_{n-1}$$xn-1,3.the third layer on the point $$x_{n-1}$$xn-1,4.the fourth layer on the point $$x_{n}$$xn and finally,5.the fifth (final) layer on the point $$x_{n+1}$$xn+1,the new algorithm has vanished the phase-lag and its first, second, third and fourth derivatives,the new algorithm has improved stability characteristics for the general problems,the new algorithm is of P-stable type since it has an interval of periodicity equal to $$\left( 0, \infty \right) $$0,∞. For the new developed algorithm we present a detailed numerical analysis (local truncation error and stability analysis). The effectiveness of the new developed algorithm is evaluated with the approximate solution of coupled differential equations arising from the Schrödinger type.