scholarly journals Interval Runge-Kutta Methods with Variable Step Sizes

2019 ◽  
Vol 25 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Andrzej Marciniak ◽  
Barbara Szyszka
Keyword(s):  
Variable Step ◽  
Runge Kutta

scholarly journals Implementing Adams Methods with Preassigned Stepsize Ratios

2010 ◽  
Vol 2010 ◽  
pp. 1-19
Author(s):  
David J. López ◽  
José G. Romay
Keyword(s):  
Numerical Study ◽  
Step Length ◽  
Step Size ◽  
Cpu Time ◽  
Numerical Tests ◽  
Function Calls ◽  
Adams Methods ◽  
Variable Step ◽  
Runge Kutta

Runge-Kutta and Adams methods are the most popular codes to solve numerically nonstiff ODEs. The Adams methods are useful to reduce the number of function calls, but they usually require more CPU time than the Runge-Kutta methods. In this work we develop a numerical study of a variable step length Adams implementation, which can only take preassigned step-size ratios. Our aim is the reduction of the CPU time of the code by means of the precalculation of some coefficients. We present several numerical tests that show the behaviour of the proposed implementation.

scholarly journals Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

Computation ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 32
Author(s):  
Musa Demba ◽  
Poom Kumam ◽  
Wiboonsak Watthayu ◽  
Pawicha Phairatchatniyom
Keyword(s):  
Error Estimation ◽  
Variable Step Size ◽  
Local Error ◽  
Step Size ◽  
Local Error Estimation ◽  
Nyström Methods ◽  
Periodic Problems ◽  
Variable Step ◽  
Exponentially Fitted ◽  
Runge Kutta

In this work, a pair of embedded explicit exponentially-fitted Runge–Kutta–Nyström methods is formulated for solving special second-order ordinary differential equations (ODEs) with periodic solutions. A variable step-size technique is used for the derivation of the 5(3) embedded pair, which provides a cheap local error estimation. The numerical results obtained signify that the new adapted method is more efficient and accurate compared with the existing methods.

A Comparative Study of Digital Integration Methods

SIMULATION ◽  
1969 ◽  
Vol 12 (2) ◽  
pp. 87-94 ◽  
Author(s):  
Hinrich R. Martens
Keyword(s):  
Linear Systems ◽  
State Transition ◽  
General Purpose ◽  
The State ◽  
Variable Step Size ◽  
Step Size ◽  
Synchronous Interaction ◽  
Variable Step ◽  
Runge Kutta ◽  
Speed Accuracy

This paper is a brief report on an investigation into the performance of integration routines used in the simula tion of dynamic systems by general-purpose digital simu lation programs. The performance is compared with respect to overall speed, accuracy, and convenience. Several routines are considered, including rectangular, Runge-Kutta, Runge-Kutta-Blum, Runge-Kutta-Mer son, and Adams-Moulton methods. The last two may use either fixed- or variable-step-size methods. The Tus tin method and the State-Transition method are also included. By examining the performance of the routines through tests on representative systems, it is shown that for the simulation of linear systems the state transition method works best. For the general simulation of linear and non linear systems the variable-step-size Runge-Kutta-Mer son method proves to be most accurate and most efficient. The variable-step-size method may be designed to gen erate outputs at uniform intervals, thus greatly enhanc ing its value when synchronous interaction with other parts of a simulation is required.

EXPONENTIALLY FITTED RUNGE–KUTTA FOURTH ALGEBRAIC ORDER METHODS FOR THE NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION AND RELATED PROBLEMS

2000 ◽  
Vol 11 (04) ◽  
pp. 785-807 ◽  
Author(s):  
P. S. WILLIAMS ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Solution ◽  
Fourth Order ◽  
Numerical Experiments ◽  
Schrodinger Equation ◽  
Variable Step ◽  
Algebraic Order ◽  
Exponentially Fitted ◽  
Runge Kutta ◽  
Step Algorithm

Fourth order exponential and trigonometric fitted Runge–Kutta methods are developed in this paper. They are applied to problems involving the Schrödinger equation and to other related problems. Numerical results show the superiority of these methods over conventional fourth order Runge–Kutta methods. Based on the methods developed in this paper, a variable-step algorithm is proposed. Numerical experiments show the efficiency of the new algorithm.

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