Optimized Three-Stage Implicit Runge-Kutta Methods for the Numerical Solution of Problems with Oscillatory Solutions

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

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Phase Fitted And Amplification Fitted Of Runge-Kutta-Fehlberg Method Of Order 4(5) For Solving Oscillatory Problems

Baghdad Science Journal ◽

10.21123/bsj.2020.17.2(si).0689 ◽

2020 ◽

Vol 17 (2(SI)) ◽

pp. 0689

Author(s):

Mohammed Salih ◽

Fudziah Ismail ◽

Norazak Senu

Keyword(s):

Periodic Solution ◽

Approximate Solution ◽

Numerical Solution ◽

Present Method ◽

Phase Lag ◽

Real Solution ◽

Oscillatory Solutions ◽

Dispersion Error ◽

Recent Method ◽

Runge Kutta

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg method.

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Optimized Two-Stage Implicit Runge-Kutta Methods for the Numerical Solution of Problems with Oscillatory Solutions

10.1063/1.3498426 ◽

2010 ◽

Author(s):

N. G. Tselios ◽

Z. A. Anastassi ◽

T. E. Simos ◽

Theodore E. Simos ◽

George Psihoyios ◽

...

Keyword(s):

Numerical Solution ◽

Two Stage ◽

Oscillatory Solutions ◽

Runge Kutta

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

International Journal of Modern Physics A ◽

10.1142/s0217751x96002169 ◽

1996 ◽

Vol 11 (26) ◽

pp. 4731-4744 ◽

Cited By ~ 4

Author(s):

T. E. SIMOS ◽

P. S. WILLIAMS

Keyword(s):

Schrödinger Equation ◽

Numerical Solution ◽

Schrodinger Equation ◽

Phase Lag ◽

Kutta Method ◽

Oscillatory Solutions ◽

New Approach ◽

Runge Kutta Method ◽

Oscillating Solutions ◽

Runge Kutta

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.

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