A New High Order Method with Optimal Stability and Phase Properties

作者：Medvedev, Maxim A.; Simos, Theodore E.^*

来源：MATCH-Communications in Mathematical and in Computer Chemistry, 2018, 79(1): 215-260.

• INITIAL-VALUE-PROBLEMS
• KUTTA-NYSTROM METHOD
• P-STABLE METHOD
• TRIGONOMETRICALLY-FITTED METHODS
• SYMMETRIC MULTISTEP METHODS
• PREDICTOR-CORRECTOR METHOD
• SCHRODINGER-EQUATION
• NUMERICAL-SOLUTION
• OSCILLATING SOLUTIONS
• 4-STEP METHODS

In the present paper we develop, for the first time in the literature, a high algebraic order P-stable symmetric two step method with eliminated phase-lag and its derivatives up to order three. The development of the new scheme is based on the following procedure: (1) The necessary and sufficient conditions for P-stability are satisfied. (2) The condition of the elimination of the phase-lag is also satisfied and finally (3) The conditions of the elimination of the derivatives of the phase-lag up to order three are also satisfied.
Based on the above the coefficients of the method is determined.
The result of the above described procedure is the construction, for the first time in the literature, of a three stages P-stable tenth algebraic order symmetric two step method with vanished phase-lag and its first, second and third derivatives.
The following investigation is also presented, for the new obtained method:
1. the construction of the scheme (based on the above procedure of three stages),
2. the computation of its local truncation error(LTE),
3. the determination of the asymptotic form of the LTE, which will be based on the radial Schrotability equation,
4. the stability analysis of the computation of the stability domain and the interval of periodicity,
5. the determination of the embedded pair for the LTE control procedure and the definition the change of the stepsize of the integration and determination of the variable step procedure,
6. the evaluation of the computational effectiveness of the new obtained pair with application on: (i) the resonance problem of the radial Schrodinger equation and on (ii) the coupled differential equations arising form the Schrodinger equation.
The above achievements leads to the conclusion that the new obtained P-stable high algebraic order scheme with vanished phase-lag and its derivatives up to order three is more efficient methods than the existed ones.

• 出版日期2018
• 单位内江师范学院