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손은성,임덕원,안종선,신미리,천세범 사단법인 항법시스템학회 2019 Journal of Positioning, Navigation, and Timing Vol.8 No.4
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.
An Error Embedded Runge-Kutta Method for Initial Value Problems
Bu, Sunyoung,Jung, WonKyu,Kim, Philsu Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.2
In this paper, we propose an error embedded Runge-Kutta method to improve the traditional embedded Runge-Kutta method. The proposed scheme can be applied into most explicit embedded Runge-Kutta methods. At each integration step, the proposed method is comprised of two equations for the solution and the error, respectively. These solution and error are obtained by solving an initial value problem whose solution has the information of the error at each integration step. The constructed algorithm controls both the error and the time step size simultaneously and possesses a good performance in the computational cost compared to the original method. For the assessment of the effectiveness, the van der Pol equation and another one having a difficulty for the global error control are numerically solved. Finally, a two-body Kepler problem is also used to assess the efficiency of the proposed algorithm.
WEAKLY STOCHASTIC RUNGE-KUTTA METHOD WITH ORDER 2
Soheili, Ali R.,Kazemi, Zahra 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.1
Many deterministic systems are described by Ordinary differential equations and can often be improved by including stochastic effects, but numerical methods for solving stochastic differential equations(SDEs) are required, and work in this area is far less advanced than for deterministic differential equations. In this paper,first we follow [7] to describe Runge-Kutta methods with order 2 from Taylor approximations in the weak sense and present two well known Runge-Kutta methods, RK2-TO and RK2-PL. Then we obtain a new 3-stage explicit Runge-Kutta with order 2 in weak sense and compare the numerical results among these three methods.
CHANDRU, M.,PONALAGUSAMY, R.,ALPHONSE, P.J.A. The Korean Society for Computational and Applied M 2017 Journal of applied mathematics & informatics Vol.35 No.1
A new fifth-order weighted Runge-Kutta algorithm based on heronian mean for solving initial value problem in ordinary differential equations is considered in this paper. Comparisons in terms of numerical accuracy and size of the stability region between new proposed Runge-Kutta(5,5) algorithm, Runge-Kutta (5,5) based on Harmonic Mean, Runge-Kutta(5,5) based on Contra Harmonic Mean and Runge-Kutta(5,5) based on Geometric Mean are carried out as well. The problems, methods and comparison criteria are specified very carefully. Numerical experiments show that the new algorithm performs better than other three methods in solving variety of initial value problems. The error analysis is discussed and stability polynomials and regions have also been presented.
M. Chandru,R. Ponalagusamy,P.J.A. ALPHONSE 한국전산응용수학회 2017 Journal of applied mathematics & informatics Vol.35 No.1
A new fth-order weighted Runge-Kutta algorithm based on heronian mean for solving initial value problem in ordinary dierential equations is considered in this paper. Comparisons in terms of numerical accuracy and size of the stability region between new proposed Runge- Kutta(5,5) algorithm, Runge-Kutta (5,5) based on Harmonic Mean, Runge- Kutta(5,5) based on Contra Harmonic Mean and Runge-Kutta(5,5) based on Geometric Mean are carried out as well. The problems, methods and comparison criteria are specied very carefully. Numerical experiments show that the new algorithm performs better than other three methods in solving variety of initial value problems. The error analysis is discussed and stability polynomials and regions have also been presented.
Weakly stochastic Runge-Kutta method with order 2
Ali R. Soheili,Zahra Kazemi 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.1
Many deterministic systems are described by Ordinary differential equations and can often be improved by including stochastic effects, but numerical methods for solving stochastic differential equations(SDEs) are required, and work in this area is far less advanced than for deterministic differential equations. In this paper,first we follow [7] to describe Runge-Kutta methods with order 2 from Taylor approximations in the weak sense and present two well known Runge-Kutta methods, RK2-TO and RK2-PL. Then we obtain a new 3-stage explicit Runge-Kutta with order 2 in weak sense and compare the numerical results among these three methods.
Numerical analysis of Legendre-Gauss-Radau and Legendre-Gauss collocation methods
Daoyong Chen,Hongjiong Tian 한국전산응용수학회 2015 Journal of applied mathematics & informatics Vol.33 No.5
In this paper, we provide numerical analysis of so-called Legendre Gauss-Radau and Legendre-Gauss collocation methods for ordinary differential equations. After recasting these collocation methods as Runge-Kutta methods, we prove that the Legendre-Gauss collocation method is equivalent to the well-known Gauss method, while the Legendre-Gauss-Radau collocation method does not belong to the classes of Radau IA or Radau IIA methods in the Runge-Kutta literature. Making use of the well-established theory of Runge-Kutta methods, we study stability and accuracy of the Legendre-Gauss-Radau collocation method. Numerical experiments are conducted to confirm our theoretical results on the accuracy and numerical stability of the Legendre-Gauss-Radau collocation method, and compare Legendre-Gauss collocation method with the Gauss method.
NUMERICAL ANALYSIS OF LEGENDRE-GAUSS-RADAU AND LEGENDRE-GAUSS COLLOCATION METHODS
CHEN, DAOYONG,TIAN, HONGJIONG The Korean Society for Computational and Applied M 2015 Journal of applied mathematics & informatics Vol.33 No.5
In this paper, we provide numerical analysis of so-called Legendre Gauss-Radau and Legendre-Gauss collocation methods for ordinary differential equations. After recasting these collocation methods as Runge-Kutta methods, we prove that the Legendre-Gauss collocation method is equivalent to the well-known Gauss method, while the Legendre-Gauss-Radau collocation method does not belong to the classes of Radau IA or Radau IIA methods in the Runge-Kutta literature. Making use of the well-established theory of Runge-Kutta methods, we study stability and accuracy of the Legendre-Gauss-Radau collocation method. Numerical experiments are conducted to confirm our theoretical results on the accuracy and numerical stability of the Legendre-Gauss-Radau collocation method, and compare Legendre-Gauss collocation method with the Gauss method.
이해균,이남주 대한토목학회 2016 KSCE Journal of Civil Engineering Vol.20 No.2
The wet-dry scheme for moving boundary treatment is implemented in the framework of discontinuous Galerkin shallow water equations. As a formulation of approximate Riemann solver, the HLL (Harten-Lax-van Leer) numerical fluxes are employed and the TVB (Total Variation Bounded) slope limiter is used for the removal of unnecessary oscillations. As benchmark test problems, the dam-break problems and the classical problem of periodic oscillation in the parabolic bowl are solved with linear triangular elements and second-order Runge-Kutta scheme. The results are compared with exact solutions and the numerical solutions of previous study. For a more practical application, the implicit Runge-Kutta scheme is employed for the bottom friction terms and the moving shoreline in a rectangular basin of varying slopes is simulated. In all case studies, good agreement is observed with exact solutions or other well-known numerical solutions.