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

G0701-4-1 Numerical Analyses by Runge Kutta with Variable Step Size for Engine Bearings

2009 ◽  
Vol 2009.3 (0) ◽  
pp. 135-136
Author(s):  
Hirotsugu HAYASHI ◽  
Yuta Komatsu
Keyword(s):  
Numerical Analyses ◽  
Variable Step Size ◽  
Step Size ◽  
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.

scholarly journals Numerical integrators for Lagrangian oceanography

2020 ◽  
Author(s):  
Tor Nordam ◽  
Rodrigo Duran
Keyword(s):  
Velocity Field ◽  
Spline Interpolation ◽  
Horizontal Resolution ◽  
Ocean Model ◽  
Computational Effort ◽  
Ocean Currents ◽  
Global Ocean ◽  
Wide Range ◽  
Variable Step ◽  
Runge Kutta

Abstract. A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field pre-calculated with an ocean model. Mathematically, this is simply numerical integration of an Ordinary Differential Equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km, and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge-Kutta integrators of orders 1–4, as well as three variable-step Runge-Kutta methods (Bogacki-Shampine 3(2), Dormand-Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, that are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand-Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. The best results are seen for coarser resolutions (4 km and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations from atmospheric models.

scholarly journals Numerical integrators for Lagrangian oceanography

2020 ◽  
Vol 13 (12) ◽  
pp. 5935-5957
Author(s):  
Tor Nordam ◽  
Rodrigo Duran
Keyword(s):  
Velocity Field ◽  
Spline Interpolation ◽  
Computational Cost ◽  
Horizontal Resolution ◽  
Ocean Model ◽  
Ocean Currents ◽  
Global Ocean ◽  
Wide Range ◽  
Variable Step ◽  
Runge Kutta

Abstract. A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field precalculated with an ocean model. Mathematically, this is simply numerical integration of an ordinary differential equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field in both space and time, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km, and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge–Kutta integrators of orders 1–4, as well as three variable-step Runge–Kutta methods (Bogacki–Shampine 3(2), Dormand–Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, which are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand–Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. Equivalently, the special-purpose integrators can provide the same accuracy as standard methods at a reduced computational cost. The best results are seen for coarser resolutions (4 and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations on data from atmospheric models.

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