EXPLICIT NUMEROV TYPE METHODS FOR SECOND ORDER IVPs WITH OSCILLATING SOLUTIONS

2001 ◽  
Vol 12 (05) ◽  
pp. 657-666 ◽  
Author(s):  
G. PAPAGEORGIOU ◽  
CH. TSITOURAS ◽  
I. TH. FAMELIS
Keyword(s):  
Numerical Methods ◽  
Taylor Expansion ◽  
High Accuracy ◽  
Order Conditions ◽  
Numerical Results ◽  
Phase Lag ◽  
New Approach ◽  
New Methods ◽  
Oscillating Solutions ◽  
Algebraic Order

New explicit hybrid Numerov type methods are presented in this paper. These efficient methods are constructed using a new approach, where we do not need the use of the intermediate high accuracy interpolatory nodes, since only the Taylor expansion of the internal points is needed. The methods share sixth algebraic order at a cost of five stages per step while their phase-lag order is 14 and partly satisfy the dissipation order conditions. It has be seen that the property of phase-lag is more important than the nonempty interval in constructing numerical methods for the solution of Schrödinger equation and related problems.1–3 Numerical results over some well known problems in physics and mechanics indicate the superiority of the new methods.

scholarly journals SPECIAL OPTIMIZED RUNGE–KUTTA METHODS FOR IVPs WITH OSCILLATING SOLUTIONS

2004 ◽  
Vol 15 (01) ◽  
pp. 1-15 ◽  
Author(s):  
Z. A. ANASTASSI ◽  
T. E. SIMOS
Keyword(s):  
Efficient Solution ◽  
Step Length ◽  
Variable Coefficients ◽  
Numerical Results ◽  
Phase Lag ◽  
Double Precision ◽  
New Methods ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
Runge Kutta

In this paper we present a family of explicit Runge–Kutta methods of 5th algebraic order, one of which has variable coefficients, for the efficient solution of problems with oscillating solutions. Emphasis is placed on the phase-lag property in order to show its importance with regards to problems with oscillating solutions. Basic theory of Runge–Kutta methods, phase-lag analysis and construction of the new methods are described. Numerical results obtained for known problems show the efficiency of the new methods when they are compared with known methods in the literature. Furthermore we note that the method with variable coefficients appears to have much higher accuracy, which gets close to double precision, when the product of the frequency with the step-length approaches certain values. These values are constant and independent of the problem solved and depend only on the method used and more specifically on the expressions used to achieve higher algebraic order.

EXPLICIT EIGHTH ORDER NUMEROV-TYPE METHODS WITH REDUCED NUMBER OF STAGES FOR OSCILLATORY IVPs

2008 ◽  
Vol 19 (06) ◽  
pp. 957-970 ◽  
Author(s):  
I. Th. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
Computational Cost ◽  
Phase Lag ◽  
Initial Value ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Algebraic Order

Using a new methodology for deriving hybrid Numerov-type schemes, we present new explicit methods for the solution of second order initial value problems with oscillating solutions. The new methods attain algebraic order eight at a cost of eight function evaluations per step which is the most economical in computational cost that can be found in the literature. The methods have high amplification and phase-lag order characteristics in order to suit to the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

A NEW MODIFIED RUNGE–KUTTA–NYSTRÖM METHOD WITH PHASE-LAG OF ORDER INFINITY FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS

2000 ◽  
Vol 11 (06) ◽  
pp. 1195-1208 ◽  
Author(s):  
T. E. SIMOS ◽  
JESUS VIGO AGUIAR
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
New Method ◽  
Phase Lag ◽  
New Approach ◽  
Nyström Methods ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
Runge Kutta

In this paper, a new approach for developing efficient Runge–Kutta–Nyström methods is introduced. This new approach is based on the requirement of annihilation of the phase-lag (i.e., the phase-lag is of order infinity) and on a modification of Runge–Kutta–Nyström methods. Based on this approach, a new modified Runge–Kutta–Nyström fourth algebraic order method is developed for the numerical solution of Schrödinger equation and related problems. The new method has phase-lag of order infinity and extended interval of periodicity. Numerical illustrations on the radial Schrödinger equation and related problems with oscillating solutions indicate that the new method is more efficient than older ones.

scholarly journals The Use of Phase Lag and Amplification Error Derivatives for the Construction of a Modified Runge-Kutta-Nyström Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
D. F. Papadopoulos ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Results ◽  
Phase Lag ◽  
Nyström Method ◽  
Nystrom Method ◽  
Modified Method ◽  
Algebraic Order ◽  
Runge Kutta

A new modified Runge-Kutta-Nyström method of fourth algebraic order is developed. The new modified RKN method is based on the fitting of the coefficients, due to the nullification not only of the phase lag and of the amplification error, but also of their derivatives. Numerical results indicate that the new modified method is much more efficient than other methods derived for solving numerically the Schrödinger equation.

High-algebraic, high-phase lag methods for accurate computations for the elastic- scattering phase shift problem

1998 ◽  
Vol 76 (6) ◽  
pp. 473-493 ◽  
Author(s):  
T E Simos
Keyword(s):  
Elastic Scattering ◽  
Phase Shift ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Finite Difference Methods ◽  
Phase Lag ◽  
New Methods ◽  
Shift Problem ◽  
Algebraic Order ◽  
Scattering Phase

A family of three new hybrid eighth-algebraic-order two-step methods with phase lag of order 16, 18, and 20 are developed for computing elastic-scattering phase shifts of the one-dimensional Schrödinger equation. Based on these new methods, we obtain some new embedded variable-step procedures for the numerical integration of the Schrödinger equation. Numerical results obtained for both the integration of the phase-shift problem for the well known case of the Lennard–Jones potential and the integration of coupled differential equation arising from the Schrödinger equation show that these new methods are better than other finite-difference methods. PACS Nos.: 02.00, 02.70, 03.00, 03.65

scholarly journals Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Kasim Hussain ◽  
Fudziah Ismail ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Order Conditions ◽  
Numerical Results ◽  
Type Method ◽  
First Order ◽  
New Methods ◽  
Runge Kutta ◽  
Fifth Order

A Runge-Kutta type method for directly solving special fourth-order ordinary differential equations (ODEs) which is denoted by RKFD method is constructed. The order conditions of RKFD method up to order five are derived; based on the order conditions, three-stage fourth- and fifth-order Runge-Kutta type methods are constructed. Zero-stability of the RKFD method is proven. Numerical results obtained are compared with the existing Runge-Kutta methods in the scientific literature after reducing the problems into a system of first-order ODEs and solving them. Numerical results are presented to illustrate the robustness and competency of the new methods in terms of accuracy and number of function evaluations.

scholarly journals New 4(3) Pairs Diagonally Implicit Runge-Kutta-Nyström Method for Periodic IVPs

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Norazak Senu ◽  
Mohamed Suleiman ◽  
Fudziah Ismail ◽  
Norihan Md Arifin
Keyword(s):  
Differential Equations ◽  
Truncation Error ◽  
Second Order ◽  
Phase Lag ◽  
Initial Value ◽  
Third Order ◽  
Second Order Differential Equations ◽  
New Methods ◽  
Oscillating Solutions ◽  
Runge Kutta

New 4(3) pairs Diagonally Implicit Runge-Kutta-Nyström (DIRKN) methods with reduced phase-lag are developed for the integration of initial value problems for second-order ordinary differential equations possessing oscillating solutions. Two DIRKN pairs which are three- and four-stage with high order of dispersion embedded with the third-order formula for the estimation of the local truncation error. These new methods are more efficient when compared with current methods of similar type and with the L-stable Runge-Kutta pair derived by Butcher and Chen (2000) for the numerical integration of second-order differential equations with periodic solutions.

NUMEROV-TYPE METHODS FOR OSCILLATORY LINEAR INITIAL VALUE PROBLEMS

2009 ◽  
Vol 20 (03) ◽  
pp. 383-398 ◽  
Author(s):  
I. TH. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
Type Method ◽  
Numerical Tests ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
High Phase

We present a new explicit Numerov-type method for the solution of second-order linear initial value problems with oscillating solutions. The new method attains algebraic order seven at a cost of six function evaluations per step. The method has the characteristic of zero dissipation and high phase-lag order making it suitable for the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

A NEW SYMMETRIC EIGHT-STEP PREDICTOR-CORRECTOR METHOD FOR THE NUMERICAL SOLUTION OF THE RADIAL SCHRÖDINGER EQUATION AND RELATED ORBITAL PROBLEMS

2011 ◽  
Vol 22 (02) ◽  
pp. 133-153 ◽  
Author(s):  
G. A. PANOPOULOS ◽  
Z. A. ANASTASSI ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
New Method ◽  
Phase Lag ◽  
Oscillating Solutions ◽  
Orbital Problems ◽  
Radial Schrödinger Equation ◽  
Algebraic Order ◽  
Predictor Corrector ◽  
Minimal Phase

A new general multistep predictor-corrector (PC) pair form is introduced for the numerical integration of second-order initial-value problems. Using this form, a new symmetric eight-step predictor-corrector method with minimal phase-lag and algebraic order ten is also constructed. The new method is based on the multistep symmetric method of Quinlan–Tremaine,1 with eight steps and 8th algebraic order and is constructed to solve numerically the radial time-independent Schrödinger equation. It can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the new method to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with minimal phase-lag is the most efficient of all the compared methods and for all the problems solved.

scholarly journals Solving Oscillating Problems Using Modifying Runge-Kutta Methods

2021 ◽  
Vol 34 (4) ◽  
pp. 58-67
Author(s):  
Zainab Khaled Ghazal ◽  
Kasim Abbas Hussain
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lag Phase ◽  
Phase Lag ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Runge Kutta ◽  
Order Four

     This paper develop conventional Runge-Kutta methods of order four and order five to solve ordinary differential equations with oscillating solutions. The new modified Runge-Kutta methods (MRK) contain the invalidation of phase lag, phase lag’s derivatives, and amplification error. Numerical tests from their outcomes show the robustness and competence of the new methods compared to the well-known Runge-Kutta methods in the scientific literature.

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