• Journal of Positioning, Navigation, and Timing
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• v.8 no.4
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• pp.201-208
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• 2019

Numerical integration is necessary for satellite orbit determination and its prediction. The numerical integration algorithm can be divided into single-step and multi-step method. There are lots of single-step and multi-step methods. However, the Runge-Kutta method in single-step and the Adams method in multi-step are generally used in global navigation satellite system (GNSS) satellite orbit. In this study, 4th and 8th order Runge-Kutta methods and various order of Adams-Bashforth-Moulton methods were used for GLObal NAvigation Satellite System (GLONASS) orbit integration using its broadcast ephemeris and these methods were compared with international GNSS service (IGS) final products for 7days. As a result, the RMSE of Runge-Kutta methods were 3.13m and 4th and 8th order Runge-Kutta results were very close and also 3rd to 9th order Adams-Bashforth-Moulton results. About result of computation time, this study showed that 4th order Runge-Kutta was the fastest. However, in case of 8th order Runge-Kutta, it was faster than 14th order Adams-Bashforth-Moulton but slower than 13th order Adams-Bashforth-Moulton in this study.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.4
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• pp.238-266
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• 2013

The Interaction Picture (IP) method is a valuable alternative to Split-step methods for solving certain types of partial differential equations such as the nonlinear Schr$\ddot{o}$dinger equation or the Gross-Pitaevskii equation. Although very similar to the Symmetric Split-step (SS) method in its inner computational structure, the IP method results from a change of unknown and therefore do not involve approximation such as the one resulting from the use of a splitting formula. In its standard form the IP method such as the SS method is used in conjunction with the classical 4th order Runge-Kutta (RK) scheme. However it appears to be relevant to look for RK scheme of higher order so as to improve the accuracy of the IP method. In this paper we investigate 5th order Embedded Runge-Kutta schemes suited to be used in conjunction with the IP method and designed to deliver a local error estimation for adaptive step size control.

• Journal of Ocean Engineering and Technology
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• v.4 no.2
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• pp.73-84
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• 1990

A circular cylinder is oscillated in th otherwise quiescent viscous fluid. Numerical analysis performed for this problem by using the fourth-order Runge-kutta method for the unsteady Navier-stokes equations. For K(Kelegan-Carpenter's No.)=5, the flow developed symmetrically, while for K=10, it revealed random patterns. The coefficient of the rms force is overestimated by 20-30% compared with the experimental result.

• 제어로봇시스템학회:학술대회논문집
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• 1991.10a
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• pp.383-388
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• 1991

This paper analizes two numerical integration methods, both based on the Runge Kutta 4-th order formula for deterministic systems, for digital simulation of a differential equation driven by white noise. It is shown that a "standard' Runge Kutta method for stochasitic systems yields solutions of Stratonovich differential equations, while Riggs and Phillips' method results in solutions of Ito differential equations. Therefore the white noise differential equation must be converted into the equivalent Ito equation before the latter method is used. Digital simulation results for a simple differential equation are also presented.nted.

• Journal of computational fluids engineering
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• v.15 no.2
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• pp.55-61
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• 2010

A dependence of weighting parameter in preconditioning method for solving low Mach number flow with incompressible flow nature is investigated. The present preconditioning method employs a finite-difference method applied Roe‘s flux difference splitting approximation with the MUSCL-TVD scheme and 4th-order Runge-Kutta method in curvilinear coordinates. From the computational results of benchmark flows through a 2-D backward-facing step duct it is confirmed that there exists a suitable value of the weighting parameter for accurate and stable computation. A useful method to determine the weighting parameter is introduced. With this method, high accuracy and stable computational results were obtained for the flow with low Mach number in the range of Mach number less than 0.3.

• Journal of Ocean Engineering and Technology
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• v.4 no.2
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• pp.223-223
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• 1990

A circular cylinder is oscillated in th otherwise quiescent viscous fluid. Numerical analysis performed for this problem by using the fourth-order Runge-kutta method for the unsteady Navier-stokes equations. For K(Kelegan-Carpenter''s No.)=5, the flow developed symmetrically, while for K=10, it revealed random patterns. The coefficient of the rms force is overestimated by 20-30% compared with the experimental result.

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
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• v.28 no.6
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• pp.573-578
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• 2010

Precise determination of satellite positions is necessary to improve positioning accuracy in GNSS. In this study, GLONASS orbits were predicted from broadcast ephemeris using the 4th-order Runge-Kutta numerical integration method and their accuracy dependence on the integration step and the integration time was analyzed. The 3D RMS (Root Mean Square) differences between the results from I-second integration step and 300-second integration step was about 3 cm, but the processing time was one hundred times less for the I-second integration time case. For trials of different integration times, the 3D RMS errors were 8.3 m, 187.3 m, and 661.5 m for 30-, 150-, and 300-minutes of integration time, respectively. Though this integration-time analysis, we concluded that the accuracy gets higher with a shorter integration time. Thus we suggest forward and backward integration methods to improve GLONASS positioning accuracy, and with this method we can achieve a 5-meter level of 3-D orbit accuracy.

• 한국전산유체공학회:학술대회논문집
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• 2005.04a
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• pp.197-201
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• 2005

According to operating conditions of each engine, a PCV valve has various flow rates and pressure characteristic. In a developed country, it has been developing by a computational design simulation. But, Korean companies have no ability of technical design for a PCV valve. So, they depend on their experiments and copy the designs of foreign companies when they need to design new PCV valves. These problems cause increase of error rate and take much time. Hence, optimal design for a PCV valve is needed to secure for continuous competition against foreign automobile companies. In this study, we used 4th order Runge-Kutta method for the prediction of spool movements and applied Bernoulli's equation for the determination of flow area. A spool geometry and spool displacement were predicted to be satisfied in comparison with their experiment.

• Journal of Auto-vehicle Safety Association
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• v.5 no.1
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• pp.25-30
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• 2013

The main purpose of this research is to predict dynamic behavior of door locking system for side impact safety and the design process to avoid door opening is introduced. The equations of motion that represent the system are obtained from the energy equation. From them, the motion of door handle is predicted by using Runge-Kutta $4^{th}$ order method and the simulation result is compared with the real crash data. Also, the design guide to define the properties of door locking system from the standpoint of avoiding door opening phenomenon is introduced.

• Applied Chemistry for Engineering
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• v.9 no.7
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• pp.1079-1084
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• 1998

Reaction kinetics between 4,4'-dihexyl methane diisocyanate($H_{12}MDI$) and n-hexanol in toluene with dibutyltin dilaurate(DBTDL) as catalyst was studied by experimental measurements and mathematical modeling. Experiments were carried out at various temperatures, catalyst concentrations and [NCO]/[OH] ratios, and the reaction kinetics were described by two second-order reactions, the one between NCO and OH leading to urethane and the other between urethane and NCO leading to allophanate. The rate constants were estimated by the Runge-Kutta 4th-order method. Experiments and mathematical simulations showed a good agreement for various experimental conditions. The [allophanate]/[urethane] ratios at 90% conversion of initial NCO were estimated to be over 20% for most conditions employed in the present study, indicating that allophanate formation might significantly affect the properties of urethane polymers.