High Phase-Lag-Order Runge--Kutta and Nyström Pairs

1999 ◽  
Vol 21 (2) ◽  
pp. 747-763 ◽  
Author(s):  
S. N. Papakostas ◽  
Ch. Tsitouras
Keyword(s):  
Phase Lag ◽  
Runge Kutta ◽  
High Phase

scholarly journals Evolutionary Derivation of Runge–Kutta Pairs of Orders 5(4) Specially Tuned for Problems with Periodic Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2306
Author(s):  
Vladislav N. Kovalnogov ◽  
Ruslan V. Fedorov ◽  
Andrey V. Chukalin ◽  
Theodore E. Simos ◽  
Charalampos Tsitouras
Keyword(s):  
Periodic Solutions ◽  
State Of The Art ◽  
Relevant Literature ◽  
Special Property ◽  
Phase Lag ◽  
The Family ◽  
Standard Set ◽  
Free Parameters ◽  
Runge Kutta ◽  
High Phase

The purpose of the present work is to construct a new Runge–Kutta pair of orders five and four to outperform the state-of-the-art in these kind of methods when addressing problems with periodic solutions. We consider the family of such pairs that the celebrated Dormand–Prince pair also belongs. The chosen family comes with coefficients that all depend on five free parameters. These latter parameters are tuned in a way to furnish a new method that performs best on a couple of oscillators. Then, we observe that this trained pair outperforms other well known methods in the relevant literature in a standard set of problems with periodic solutions. This is remarkable since no special property holds such as high phase-lag order or an extended interval of periodicity.

High phase-lag order Runge Kutta pairs of orders 8(7)

2017 ◽  
Author(s):  
Ch. Tsitouras ◽  
Ioannis Th. Famelis
Keyword(s):  
Phase Lag ◽  
Runge Kutta ◽  
High Phase

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.

Dissipative high phase-lag order methods

2001 ◽  
Vol 117 (1) ◽  
pp. 35-43 ◽  
Author(s):  
Ch. Tsitouras
Keyword(s):  
Phase Lag ◽  
High Phase

A NEW METHODOLOGY FOR THE CONSTRUCTION OF OPTIMIZED RUNGE–KUTTA–NYSTRÖM METHODS

2011 ◽  
Vol 22 (06) ◽  
pp. 623-634 ◽  
Author(s):  
D. F. PAPADOPOULOS ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
First Integrals ◽  
New Method ◽  
Phase Lag ◽  
Nyström Methods ◽  
Algebraic Order ◽  
Runge Kutta ◽  
First Time ◽  
Zero Phase

In this paper, a new Runge–Kutta–Nyström method of fourth algebraic order is developed. The new method has zero phase-lag, zero amplification error and zero first integrals of the previous properties. Numerical results indicate that the new method is very efficient for solving numerically the Schrödinger equation. We note that for the first time in the literature we use the requirement of vanishing the first integrals of phase-lag and amplification error in the construction of efficient methods for the numerical solution of the Schrödinger equation.

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.

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