SOME MODIFIED RUNGE–KUTTA METHODS FOR THE NUMERICAL SOLUTION OF SOME SPECIFIC SCHRÖDINGER EQUATIONS AND RELATED PROBLEMS

Abstract

Some new modified Runge–Kutta methods with minimal phase lag are developed for the numerical solution of the eigenvalue Schrödinger equation and related problems with oscillating solutions. These methods are based on the very well-known Runge–Kutta method of order 4. For the numerical solution of the eigenvalue Schrödinger equation, we investigate two cases: (i) the specific case in which the potential V(x) is an even function with respect to x; it is assumed, also, that the wave functions tend to zero for x → ±∞; (ii) the general case for the well-known cases of the Morse potential and Woods–Saxon or optical potential. Also, we have applied the new methods to some well-known problems with oscillatory solutions. Numerical and theoretical results show that this new approach is more efficient than the well-known classical fourth order Runge–Kutta method and the Numerov method.